%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{23}
\usepackage{enumitem}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

% \usepackage{amssymb}
% \usepackage{amsmath}
% \usepackage{amsthm}
\usepackage{graphicx}
% \usepackage{enumerate}
\usepackage{tikz}
\usepackage{cleveref}
%%%\usepackage{MnSymbol} % package with disjoint union symbol \cupdot or \bigcupdot
\usetikzlibrary{arrows}
\usetikzlibrary{arrows.meta}
\tikzset{>={Latex[width=1.2mm,length=1.7mm]}}
\usepackage{tabularx}
%\usepackage{mathdots}
\usepackage{mathrsfs}
%\usepackage{tabmac_avi}
%\usepackage{ytableau}
\newenvironment{proofidea}{\noindent \paragraph{\emph{Proof idea.}}}{\hfill $\Box$}
\newenvironment{proofomitted}{\noindent \paragraph{\emph{Proof omitted.}}}{\hfill $\Box$} 

%The following set of commands is cut and paste from
%the macros file of my dissertations. Much can be
%removed most likely.
\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{obs}[thm]{Observation}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{prob}[thm]{Problem}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}%[section]
\newtheorem{alg}[thm]{Algorithm}
\newtheorem{lemma}[thm]{Lemma}
%\newtheorem{ex}[defn]{Example}
%\newtheorem{ex}[thm]{Example}
%\newtheorem{defn}[thm]{Definition}
%\newtheorem{quest}[defn]{Question}
%\newtheorem{quest}[thm]{Question}

%\newtheorem{quest}{Question}[section]
%\newtheorem{prob}[thm]{Problem}
%\newtheorem{example}[thm]{Example}

\numberwithin{equation}{section}

\newcommand{\sn}{\mathfrak{S}_n}
\newcommand{\sr}{\mathfrak{S}_r}
\newcommand{\mfs}[1]{\mathfrak{S}_{#1}}
\newcommand{\snn}{\mfs{\smash{[\ol n, n]}}}
\newcommand{\mfsmin}[1]{\mathfrak{S}_-^{#1}}
\newcommand{\bn}{\mathfrak{B}_n}
\newcommand{\br}{\mathfrak{B}_r}
\newcommand{\mfb}[1]{\mathfrak{B}_{#1}}
\newcommand{\mfbmin}[1]{\mathfrak{B}_-^{#1}}
\newcommand{\zsn}{\mathbb{Z}[\sn]}
\newcommand{\qsn}{\mathbb{Q}[\sn]}
\newcommand{\zbn}{\mathbb{Z}[\bn]}
\newcommand{\qbn}{\mathbb{Q}[\bn]}
\newcommand{\cx}{\mathbb{C}[x]}
\newcommand{\zx}{\mathbb{Z}[x]}
\newcommand{\cxn}{\mathbb{C}[x_{1,1},\dotsc,x_{n,n}]}
\newcommand{\zxn}{\mathbb{Z}[x_{1,1},\dotsc,x_{n,n}]}
\newcommand{\cqq}{\mathbb{C}[\qp12, \qm12]}
\newcommand{\zqq}{\mathbb{Z}[\qp12, \qm12]}
\newcommand{\zqQ}{\mathbb{Z}[\qp12, \qm12, \Qp12, \Qm12]}
\newcommand{\nqq}{\mathbb{N}[\qp12, \qm12]}
\newcommand{\czn}{\cqq \langle z_{1,1},\dotsc,z_{n,n} \rangle}
%\newcommand{\cz}{\cqq \langle z \rangle}
\newcommand{\cz}{\mathbb{C}[z]}
\newcommand{\zz}{\mathbb{Z}[z]}
\newcommand{\cyr}{\mathbb{C}[y_{1,1},\dotsc,x_{r,r}]}
\newcommand{\csn}{\mathbb{C}[\sn]}
\newcommand{\bitab}{\mathcal B}
\newcommand{\tab}{\mathcal U}
%\newcommand{\bctab}{\mathcal U^*}
\newcommand{\bctab}{\mathcal U^{\msfBC}}
\newcommand{\atab}{\mathcal U^{\msfA}}

\newcommand{\Ie}{{I, \emptyset}}
\newcommand{\eJ}{{\emptyset, J}}
\newcommand{\Je}{{J, \emptyset}}
\newcommand{\eI}{{\emptyset, I}}
\newcommand{\IIJJ}{{I, J}}

\newcommand{\wo}{w_0}
\newcommand{\woi}{w_0^I}
\newcommand{\woj}{w_0^J}
\newcommand{\wok}{w_0^K}
\newcommand{\wokp}{w_0^{K'}}

\newcommand{\etu}{\epsilon_{t,u}}
\newcommand{\etv}{\epsilon_{t,v}}
\newcommand{\etz}{\epsilon_{t,z}}
\newcommand{\eut}{\epsilon_{u,t}}
\newcommand{\euv}{\epsilon_{u,v}}
\newcommand{\euw}{\epsilon_{u,w}}
\newcommand{\euy}{\epsilon_{u,y}}
\newcommand{\euz}{\epsilon_{u,z}}
\newcommand{\evu}{\epsilon_{v,u}}
\newcommand{\evw}{\epsilon_{v,w}}
\newcommand{\evy}{\epsilon_{v,y}}
\newcommand{\evz}{\epsilon_{v,z}}
\newcommand{\ewy}{\epsilon_{w,y}}
\newcommand{\ewz}{\epsilon_{w,z}}
\newcommand{\eyz}{\epsilon_{y,z}}
\newcommand{\ezu}{\epsilon_{z,u}}
\newcommand{\ezw}{\epsilon_{z,w}}
\newcommand{\ezy}{\epsilon_{z,y}}
\newcommand{\eeu}{\epsilon_{e,u}}
\newcommand{\eev}{\epsilon_{e,v}}
\newcommand{\eew}{\epsilon_{e,w}}

\newcommand{\qp}[2]{q^{\frac{#1}{#2}}}
\newcommand{\qm}[2]{q^{\negthinspace\Bar\,\frac{#1}{#2}}}
\newcommand{\Qp}[2]{Q^{\frac{#1}{#2}}}
\newcommand{\Qm}[2]{Q^{\negthinspace\Bar\,\frac{#1}{#2}}}
\newcommand{\qtu}{q_{t,u}}
\newcommand{\qtv}{q_{t,v}}
\newcommand{\qtz}{q_{t,z}}
\newcommand{\quv}{q_{u,v}}
\newcommand{\quw}{q_{u,w}}
\newcommand{\quy}{q_{u,y}}
\newcommand{\qvu}{q_{v,u}}
\newcommand{\qvw}{q_{v,w}}
\newcommand{\qvy}{q_{v,y}}
\newcommand{\qvz}{q_{v,z}}
\newcommand{\qyw}{q_{y,w}}
\newcommand{\qwy}{q_{w,y}}
\newcommand{\qwz}{q_{w,z}}
\newcommand{\qzw}{q_{z,w}}
%\newcommand{\qeu}{q_{e,u}}
\newcommand{\qeu}{q_u}
\newcommand{\qet}{q_t}
\newcommand{\qey}{q_y}
%\newcommand{\qev}{q_{e,v}}
\newcommand{\qev}{q_v}
%\newcommand{\qew}{q_w}
\newcommand{\qew}{\smash{\qp{\ell(w)}2}}
\newcommand{\qitu}{q_{t,u}^{-1}}
\newcommand{\qitv}{q_{t,v}^{-1}}
\newcommand{\qitw}{q_{t,w}^{-1}}
\newcommand{\qity}{q_{t,y}^{-1}}
\newcommand{\qiut}{q_{u,t}^{-1}}
\newcommand{\qiuv}{q_{u,v}^{-1}}
\newcommand{\qiuy}{q_{u,y}^{-1}}
\newcommand{\qiuw}{q_{u,w}^{-1}}
\newcommand{\qiuz}{q_{u,z}^{-1}}
\newcommand{\qivu}{q_{v,u}^{-1}}
\newcommand{\qivw}{q_{v,w}^{-1}}
\newcommand{\qivy}{q_{v,y}^{-1}}
\newcommand{\qivz}{q_{v,z}^{-1}}
\newcommand{\qiyw}{q_{y,w}^{-1}}
\newcommand{\qiyz}{q_{y,z}^{-1}}
\newcommand{\qizw}{q_{z,w}^{-1}}
\newcommand{\qizy}{q_{z,y}^{-1}}
%\newcommand{\qiet}{q_{e,t}^{-1}}
%\newcommand{\qieu}{q_{e,u}^{-1}}
%\newcommand{\qiev}{q_{e,v}^{-1}}
%\newcommand{\qiew}{q_{e,w}^{-1}}
\newcommand{\qiew}{\qm{\ell(w)}2}
%\newcommand{\qiew}{q_w^{-1}}
\newcommand{\qiey}{q_y^{-1}}
\newcommand{\qiev}{q_v^{-1}}
%\newcommand{\qiey}{q_{e,y}^{-1}}
\newcommand{\qieyp}{q_{y'}^{-1}}
\newcommand{\qiez}{q_{e,z}^{-1}}
\newcommand{\qdiff}{\qp12 - \qm12}

\newcommand{\wT}{\widetilde T} 
\newcommand{\wTp}{\widetilde T'}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\wR}[1]{\widetilde R_{#1}}
\newcommand{\wS}[1]{\widetilde S_{#1}} 
\newcommand{\wSp}{\widetilde S'}

\newcommand{\hnq}{H_n(q)}
\newcommand{\hanq}{H_n^{\msfA}(q)}
\newcommand{\hbnq}{H_n^{\msfBC}(q)}
\newcommand{\hbjq}{H_J^{\msfBC}}
%\newcommand{\hbjq}{H_J^{\msfBC}(q)}
%\newcommand{\hbnq}{H_n^{\mathfrak{B}}(q)}
\newcommand{\hbq}[1]{H_{#1}^{\msfBC}(q)}
\newcommand{\hbnQ}{H_n^C(q,Q)}
\newcommand{\hrq}{H_r(q)}
\newcommand{\hlq}{H_\lambda(q)}
%\newcommand{\anq}{\mathcal{A}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\anq}{\mathcal{A}(n,q)}
\newcommand{\atwonq}{\mathcal{A}(2n,q)}
\newcommand{\arq}{\mathcal{A}(r;q)}
%\newcommand{\arnq}{\mathcal{A}_r(n;q)}
\newcommand{\arnq}{\mathcal{A}_r}
\newcommand{\annq}{\mathcal{A}_n(n;q)}
%\newcommand{\annnq}{\mathcal{A}_{[n],[n]}(n;q)}
\newcommand{\Ann}{\mathcal{A}_{[n],[n]}}
\newcommand{\annnq}{\mathcal{A}_{[n],[n]}}
\newcommand{\anmnq}{\mathcal{A}_{[n],M}(n;q)}
\newcommand{\amnnq}{\mathcal{A}_{M,[n]}(n;q)}
\newcommand{\alnnq}{\mathcal{A}_{L,[n]}(n;q)}
%\newcommand{\almnq}{\mathcal{A}_{L,M}(n;q)}
\newcommand{\AnM}{\mathcal{A}_{[n],M}}
\newcommand{\ALM}{\mathcal{A}_{L,M}}
\newcommand{\amlnq}{\mathcal{A}_{M,L}(n;q)}
\newcommand{\arrrq}{\mathcal{A}_{[r],[r]}(r;q)}
\newcommand{\alrrq}{\mathcal{A}_{L,[r]}(r;q)}
\newcommand{\armrq}{\mathcal{A}_{[r],M}(r;q)}
\newcommand{\trsp}{\mathcal{T}}
\newcommand{\btrsp}{\mathcal{T}^{BC}}

\newcommand{\coh}{\mathrm{H}}
\newcommand{\icoh}{\mathrm{IH}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\Hpie}{H'_{\smash{\Ie}}} 
\newcommand{\Hie}{H_\Ie} 
\newcommand{\Hpej}{H'_{\smash\eJ}} 
\newcommand{\Hej}{H_\eJ} 
\newcommand{\Hpje}{H'_\Je} 
\newcommand{\Hje}{H_\Je} 
\newcommand{\Hpei}{H'_\eI} 
\newcommand{\Hei}{H_\eI} 
\newcommand{\Hij}{H_{I,J}} 
\newcommand{\Hpij}{H'_{I,J}} 

\newcommand{\Wiep}{W^{I,\emptyset}_+}
\newcommand{\Wiem}{W^{I,\emptyset}_-}
\newcommand{\WIkpem}{(W_I)^{K',\emptyset}_-}
\newcommand{\WIekpm}{(W_I)^{\emptyset,K'}_-}
\newcommand{\WIekpp}{(W_I)^{\emptyset,K'}_+}
\newcommand{\WIenpm}{(W_I)^{\emptyset,N'}_-}
\newcommand{\WJiem}{(W_J)^{I,\emptyset}_-}
\newcommand{\WJkem}{(W_J)^{K,\emptyset}_-}
\newcommand{\WJkep}{(W_J)^{K,\emptyset}_+}
\newcommand{\WJnem}{(W_J)^{N,\emptyset}_-}
\newcommand{\Wjep}{W^{J,\emptyset}_+}
\newcommand{\Wjem}{W^{J,\emptyset}_-}
\newcommand{\Wkep}{W^{K,\emptyset}_+}
\newcommand{\Wkem}{W^{K,\emptyset}_-}
\newcommand{\Weip}{W^{\emptyset,I}_+}
\newcommand{\Weim}{W^{\emptyset,I}_-}
\newcommand{\WJeim}{(W_J)^{\emptyset,I}_-}
\newcommand{\WJekm}{(W_J)^{\emptyset,K}_-}
\newcommand{\WJenm}{(W_J)^{\emptyset,N}_-}
\newcommand{\Wejp}{W^{\emptyset,J}_+}
%\newcommand{\Wejm}{W^{\emptyset,J}_-}
\newcommand{\Wejm}{W^{J}_-}
\newcommand{\Wekp}{W^{\emptyset,K}_+}
\newcommand{\Wekpp}{W^{\emptyset,K'}_+}
\newcommand{\Wekm}{W^{\emptyset,K}_-}
\newcommand{\Wekpm}{W^{\emptyset,K'}_-}
\newcommand{\Wijp}{W^{I,J}_+}
\newcommand{\Wijm}{W^{I,J}_-}

\newcommand{\Rie}[1]{R_{#1}^{I,\emptyset}}
\newcommand{\Rpie}[1]{R^{\prime\,I,\emptyset}_{#1}}
\newcommand{\Rej}[1]{R_{#1}^{\emptyset,J}}
\newcommand{\Rpej}[1]{R^{\prime\,\emptyset,J}_{#1}}
\newcommand{\Rij}[1]{R_{#1}^{I,J}}
\newcommand{\Rpij}[1]{R^{\prime\,I,J}_{#1}}
\newcommand{\wRie}[1]{\wR_{#1}^{I,\emptyset}}
\newcommand{\wRpie}[1]{\wR^{\prime\,I,\emptyset}_{#1}}
\newcommand{\wRej}[1]{\wR_{#1}^{\emptyset,J}}
\newcommand{\wRpej}[1]{\wR^{\prime\,\emptyset,J}_{#1}}
\newcommand{\wRij}[1]{\wR_{#1}^{I,J}}
\newcommand{\wRpij}[1]{\wR^{\prime\,I,J}_{#1}}
\newcommand{\Sie}[1]{S_{#1}^{I,\emptyset}}
\newcommand{\Spie}[1]{S^{\prime\,I,\emptyset}_{#1}}
\newcommand{\Sej}[1]{S_{#1}^{\emptyset,J}}
\newcommand{\Spej}[1]{S^{\prime\,\emptyset,J}_{#1}}
\newcommand{\Sij}[1]{S_{#1}^{I,J}}
\newcommand{\Spij}[1]{S^{\prime\,I,J}_{#1}}
\newcommand{\wSie}[1]{\wS_{#1}^{I,\emptyset}}
\newcommand{\wSpie}[1]{\wS^{\prime\,I,\emptyset}_{#1}}
\newcommand{\wSej}[1]{\wS_{#1}^{\emptyset,J}}
\newcommand{\wSpej}[1]{\wS^{\prime\,\emptyset,J}_{#1}}
\newcommand{\wSij}[1]{\wS_{#1}^{I,J}}
\newcommand{\wSpij}[1]{\wS^{\prime\,I,J}_{#1}}
\newcommand{\Qie}[1]{Q_{#1}^{I,\emptyset}}
\newcommand{\Qej}[1]{Q_{#1}^{\emptyset,J}}
\newcommand{\Qij}[1]{Q_{#1}^{I,J}}
\newcommand{\pie}[1]{p_{#1}^{I,\emptyset}}
\newcommand{\pje}[1]{p_{#1}^{J,\emptyset}}
\newcommand{\pei}[1]{p_{#1}^{\emptyset,I}}
\newcommand{\pej}[1]{p_{#1}^{\emptyset,J}}
\newcommand{\pij}[1]{p_{#1}^{I\ntnsp,J}}
\newcommand{\pji}[1]{p_{#1}^{J\ntnsp,I}}
\newcommand{\rie}[1]{r_{#1}^{I\ntnsp,\emptyset}}
\newcommand{\rje}[1]{r_{#1}^{J\ntnsp,\emptyset}}
\newcommand{\rei}[1]{r_{#1}^{\emptyset,I}}
\newcommand{\rej}[1]{r_{#1}^{\emptyset,J}}
\newcommand{\rij}[1]{r_{#1}^{I\ntnsp,J}}
\newcommand{\rji}[1]{r_{#1}^{J\ntnsp,I}}
\newcommand{\redexp}[2]{s_{i_{#1}} \ntnsp \cdots s_{a_{#2}}}
\newcommand{\wtc}[2]{\widetilde{C}_{#1}(#2)}
%\newcommand{\btc}[2]{C_{#1}^{\mathfrak B}(#2)}
\newcommand{\btc}[2]{\widetilde{C}_{#1}^{\msfBC}(#2)}
\newcommand{\ewtc}[2]{(T_e + T_{#1})}

\newcommand{\imm}[1]{\mathrm{Imm}_{#1}}
\newcommand{\immlm}[1]{\mathrm{Imm}_{#1}^{L,M}}
\newcommand{\immln}[1]{\mathrm{Imm}_{#1}^{L,[n]}}
\newcommand{\immlr}[1]{\mathrm{Imm}_{#1}^{L,[r]}}
\newcommand{\immnm}[1]{\mathrm{Imm}_{#1}^{[n],M}}
\newcommand{\immrm}[1]{\mathrm{Imm}_{#1}^{[r],M}}
%\newcommand{\cimm}[1]{\text{$C$-}\mathrm{Imm}_{#1}}
\newcommand{\bnimm}[1]{\mathrm{Imm}_{#1}^{\bn}}
\newcommand{\simm}[2]{\mathrm{Imm}_{#2}^{\smash{\mfs{#1}}}}

\newcommand{\sumsb}[1]{\sum_{\substack{#1}}}  %%%for multi-line sums
\newcommand{\prodsb}[1]{\prod_{\substack{#1}}}  %%%for multi-line products
\newcommand{\inv}{\textsc{inv}}
\newcommand{\rinv}{\textsc{rinv}}
%\newcommand{\defeq}{\underset{\mathrm{def}}=}
\newcommand{\defeq}{:=} 
%\newcommand{\dfct}{\textsc{dfct}}
\newcommand{\dfct}{\mathrm{dfct}}
\newcommand{\dfctbc}{\mathrm{dfct}^{\msfBC}}
\newcommand{\dnc}{\textsc{dnc}}
\newcommand{\dc}{\textsc{dc}}
\newcommand{\pnc}{\textsc{pnc}}
\newcommand{\trunc}{\mathrm{trunc}}
\newcommand{\suff}{\mathrm{suff}}
\newcommand{\sct}{\mathrm{sct}}
\newcommand{\des}{\mathrm{des}}
\newcommand{\exc}{\mathrm{exc}}
%\newcommand{\ct}{\mathrm{ctype}}
\newcommand{\stat}{\mathrm{stat}}
\newcommand{\frobch}{\mathrm{ch}}
\newcommand{\npfrobch}{\mathrm{nch}}
\newcommand{\pfrobch}{\mathrm{pch}}
\newcommand{\spn}{\mathrm{span}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\wgt}{\mathrm{wgt}}
\newcommand{\src}{\mathrm{src}}
\newcommand{\snk}{\mathrm{snk}}
\newcommand{\type}{\mathrm{type}}
\newcommand{\ctype}{\mathrm{ctype}}
\newcommand{\inc}{\mathrm{inc}}
\newcommand{\ngr}{\mathrm{ngr}}
\newcommand{\incross}{\textsc{invnc}}
\newcommand{\cdncross}{\textsc{cdnc}}
\newcommand{\cross}{\textsc{c}}
\newcommand{\pavoiding}{$3412$-avoiding, $4231$-avoiding }
\newcommand{\avoidsp}{avoids the patterns $3412$ and $4231${}}
\newcommand{\avoidp}{avoid the patterns $3412$ and $4231${}}
\newcommand{\avoidingp}{avoiding the patterns $3412$ and $4231${}}
\newcommand{\theps}{the patterns $3412$ and $4231${}}
\newcommand{\avoidsignedp}{avoid the signed patterns $1\ol2$, $\ol21$, $\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\avoidssignedp}{avoids the signed patterns $1\ol2$,  $\ol21$, $\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\avoidingsignedp}{avoiding the signed patterns $1\ol2$, $\ol21$, $\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\ul}[1]{\underline{#1}}  
\newcommand{\ssec}[1]{\subsection{#1}{$\negthinspace$}}

\newcommand{\tr}{{\negthickspace \top \negthickspace}}
\newcommand{\ntnsp}{\negthinspace}
\newcommand{\ntksp}{\negthickspace}
\newcommand{\nTksp}{\negthickspace\negthickspace}
\newcommand{\nTknsp}{\negthickspace\negthickspace\negthinspace}
\newcommand{\nTtksp}{\negthickspace\negthickspace\negthickspace}
\newcommand{\nTTksp}{\negthickspace\negthickspace\negthickspace
	\negthickspace}
\newcommand{\nTTtksp}{\negthickspace\negthickspace\negthickspace
	\negthickspace\negthickspace}
\newcommand{\bp}{\begin{prob}}
\newcommand{\ep}{\end{prob}}


\newcommand{\kbar}[2]{\varphi_#1(#2)}

\newcommand{\hnqq}{H_n(q_2,q_4)}
\newcommand{\anqq}{\mathcal{A}(n;q_2,q_4)}
\newcommand{\arnqq}{\mathcal{A}_r(n;q_2,q_4)}
\newcommand{\almnqq}{\mathcal{A}_{L,M}(n;q_2,q_4)}
\newcommand{\annnqq}{\mathcal{A}_{[n],[n]}(n;q_2,q_4)}
\newcommand{\aT}{\widetilde{\mathcal{T}}}
\newcommand{\what}[1]{\widehat{#1}}
\newcommand{\q}[3]{q_{#1,#2,#3}}
\newcommand{\qi}[3]{q^{-1}_{#1,#2,#3}}
\newcommand{\qsum}{q_2 + q_4}
\newcommand{\qprod}{-q_2q_4}
\newcommand{\qprodi}{(-q_2q_4)^{-1}}

%\newcommand{\oglnc}{{\mathcal O}(GL(n,\mathbb{C}))}
%\newcommand{\oqglnc}{{\mathcal O}_q(GL(n,\mathbb{C}))}
%\newcommand{\oslnc}{{\mathcal O}(SL(n,\mathbb{C}))}
%\newcommand{\oslthreec}{{\mathcal O}(SL(3,\mathbb{C}))}
%\newcommand{\oqslnc}{{\mathcal O}_q(SL(n,\mathbb{C}))}
\newcommand{\oglnc}{{\mathcal O}(\mathrm{GL}_n (\mathbb C))}
\newcommand{\oqglnc}{{\mathcal O}_q(\mathrm{GL}_n (\mathbb C))}
\newcommand{\oslnc}{{\mathcal O}(\mathrm{SL}_n) (\mathbb C)}
\newcommand{\oslthreec}{{\mathcal O}(\mathrm{SL}_3 (\mathbb C))}
\newcommand{\oqslnc}{{\mathcal O}_q(\mathrm{SL}_n (\mathbb C))}
\newcommand{\ospnc}{{\mathcal O}(\mathrm{SP}_{2n} (\mathbb C))}
\newcommand{\oqspnc}{{\mathcal O}_q(\mathrm{SP}_{2n} (\mathbb C))}
\newcommand{\uglnc}{U(\mathfrak{gl}(n,\mathbb{C}))}
\newcommand{\uqglnc}{U_q(\mathfrak{gl}(n,\mathbb{C}))}
\newcommand{\uslnc}{U(\mathfrak{sl}(n,\mathbb{C}))}
\newcommand{\uqslnc}{U_q(\mathfrak{sl}(n,\mathbb{C}))}
\newcommand{\uspnc}{U(\mathfrak{sp}_{2n}(\mathbb{C}))}
\newcommand{\uqspnc}{U_q(\mathfrak{sp}_{2n}(\mathbb{C}))}
\newcommand{\mat}[1]{\mathrm{Mat}_{#1 \times #1}}
\newcommand{\spc}[1]{\mathfrak{sp}_{#1}(\mathbb{C})}
\newcommand{\slc}[1]{\mathfrak{sl}_{#1}(\mathbb{C})}
\newcommand{\fl}[1]{\mathcal{F}_{#1}}
%\newcommand{\hess}[1]{\mathcal{H}({\mathbf #1})}
\newcommand{\PiBC}{\Pi^{\msfBC}}


\newcommand{\qdet}{\mathrm{det}_q}
\newcommand{\qper}{\mathrm{per}_q}
\newcommand{\perm}{\mathrm{per}}
\newcommand{\rnk}{\mathrm{rank}}
\newcommand{\sort}{\mathrm{sort}}
\newcommand{\shape}{\mathrm{shape}}
\newcommand{\hess}{\mathrm{Hess}}
\newcommand{\ahess}{\mathrm{Hess^{\msfA}}}
\newcommand{\bhess}{\mathrm{Hess}^{\msfBB}}
\newcommand{\chess}{\mathrm{Hess}^{\msfC}}
%\newcommand{\exp}{\mathrm{exp}}
\newcommand{\permmon}[2]{#1_{1,#2_1} \ntnsp\cdots #1_{n,#2_n}}
\newcommand{\bpermmon}[2]
           {#1_{\smash{\ol n,#2_{\ol n}}} \ntnsp\cdots #1_{\smash{\ol 1,#2_{\ol 1}}}
           #1_{1,#2_{1}} \ntnsp\cdots #1_{n,#2_n} }
\newcommand{\sprod}[2]{s_{#1_1} \ntnsp\cdots s_{#1_#2}}
\newcommand{\sintprod}[3]{s_{[#1_1,#2_1]} \ntnsp\cdots s_{[#1_#3, #2_#3]}}
\newcommand{\ssm}{\smallsetminus}
\newcommand{\multiu}{\Cup}
\newcommand{\bs}{\backslash}  
\newcommand{\bfx}{\mathbf x}
\newcommand{\wleq}{\leq_W}
\newcommand{\wless}{<_W}  
\newcommand{\slambda}{\mathfrak{S}_\lambda}
\newcommand{\slambdamin}{\mathfrak{S}_\lambda^{-}}
\newcommand{\phmin}{\phantom{-}}
%\newcommand{\circd}[1]{\raisebox{.5pt}{\textcircled{\raisebox{-.9pt}{#1}}}}
\newcommand{\circd}[1]{\raisebox{-9pt}{\textcircled{\raisebox{-.9pt}{#1}}}}
%\newcommand{\oglnc}{{\mathcal O}(GL(n,\mathbb{C}))}
%\newcommand{\oqglnc}{{\mathcal O}_q(GL(n,\mathbb{C}))}
%\newcommand{\oslnc}{{\mathcal O}(SL(n,\mathbb{C}))}
%\newcommand{\oqslnc}{{\mathcal O}_q(SL(n,\mathbb{C}))}
\newcommand{\sdot}{\left[ \begin{smallmatrix} 0 & -1 \\ 1 & \phantom{-}0 \end{smallmatrix}\right]}
\newcommand{\sddot}{\left[ \begin{smallmatrix} \phantom{-}0 & 1 \\ -1 & 0 \end{smallmatrix}\right]}
\newcommand{\sreg}{\left[ \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]}
%\newcommand{\smallj}{\left[ \begin{smallmatrix} 0 & & 1 \\  & \text{\raisebox{1mm}{$\iddots$}} & \\ 1 & & 0 \end{smallmatrix}\right]}
\newcommand{\smallj}{\left[ \begin{smallmatrix} 0 & \phantom{\int} & 1 \\ & \smash{\iddots} & \phantom{T} \\ 1 & & 0 \end{smallmatrix}\right]}
\newcommand{\abdiag}[1]{\left[ \begin{smallmatrix} 1 & #1 \\ 0 & 1 \end{smallmatrix}\right]}
\newcommand{\bediag}[1]{\left[ \begin{smallmatrix} 1 & 0 \\ #1 & 1 \end{smallmatrix}\right]}
\newcommand{\upparrow}{\big \uparrow \nTksp \phantom{\uparrow}}
\newcommand{\upparrowa}{\big \uparrow \ntksp \phantom{\uparrow}}
\newcommand{\precdot}{\prec \negthickspace \negthinspace \cdot\ }
%\newcommand{\net}[2]{\mathcal F^{\mathsf{#1}}(#2)}
\newcommand{\net}[2]{\mathcal F^{\mathsf{#1}}_{#2}}
\newcommand{\snet}[2]{\mathcal S^{\mathsf{#1}}(#2)}
\newcommand{\znet}[2]{\mathcal S^{\mathsf{#1}}_{\mathrm Z}(#2)}
\newcommand{\dnet}[2]{\mathcal S^{\mathsf{#1}}_{\mathrm D}(#2)}
\newcommand{\sfA}{\textsf{A}}
\newcommand{\msfA}{\mathsf{A}}
\newcommand{\sfBB}{\textsf{B}}
\newcommand{\msfBB}{\mathsf{B}}
\newcommand{\sfC}{\textsf{C}}
\newcommand{\msfC}{\mathsf{C}}
\newcommand{\sfBC}{\textsf{BC}}
\newcommand{\msfBC}{\mathsf{BC}}
\newcommand{\YBCq}{Y_q^\mathsf{BC}}
\newcommand{\msfD}{\mathsf{D}}
\newcommand{\vertex}[1]{\mathsf{#1}}
%\renewcommand{\st}{^{\text{st}}}
%\renewcommand{\nd}{^{\text{nd}}}
%\renewcommand{\rd}{^{\text{rd}}}
\renewcommand{\th}{^{\text{th}}}
\newcommand{\phsum}{\phantom{\sum_A^Z}}
\newcommand{\phm}{\phantom M}
\newcommand{\phn}{\phantom{ni}}
\newcommand{\bcao}[1]{\mathcal{O}^{\msfBC}({#1})}
%\newcommand{\schub}[1]{\ol{\Omega}_{#1}}
\newcommand{\schub}[1]{\Omega_{#1}}
\newcommand{\Aschub}[1]{\Omega_{#1}^{\msfA}}
\newcommand{\Bschub}[1]{\Omega_{#1}^{\msfBB}}
\newcommand{\Cschub}[1]{\Omega_{#1}^{\msfC}}

\def\hsp{\def\baselinestretch{0.75}\large\normalsize}
\def\hhsp{\def\baselinestretch{0.25}\large\normalsize}
\def\hhhsp{\def\baselinestretch{0.125}\large\normalsize}
%\def\hhhpsp{\def\baselinestretch{0.13}\large\normalsize}
\def\hhhpsp{\def\baselinestretch{0.15}\large\normalsize}
\def\ssp{\def\baselinestretch{1.0}\large\normalsize}
\def\dsp{\def\baselinestretch{1.4}\large\normalsize}    % for pretty dsp
\def\tsp{\def\baselinestretch{2.0}\large\normalsize}    % for "real" dsp
\def\qsp{\def\baselinestretch{2.4}\large\normalsize}    % for bigger dsp



% \newtheorem{thm}{Theorem}
% \newtheorem{lem}{Lemma}

% \usepackage{lipsum}

%% define your title in the usual way
\title[Generalization of Deodhar's defect statistic]{A generalization of Deodhar's defect statistic for Iwahori--Hecke algebras of 
%types $\msfA$ and 
type \texorpdfstring{$\msfBC$}{BC}}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Hobbs, Parisi, Skandera, Wang]{\texorpdfstring{Gavin Hobbs\addressmark{1}, Tommy Parisi\addressmark{1},  Mark Skandera\addressmark{1}, \and Jiayuan Wang\addressmark{1}
}{Gavin Hobbs, Tommy Parisi,  Mark Skandera, and Jiayuan Wang
}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Lehigh University, Bethlehem, PA, USA}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
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\abstract{Let $H$ be the Iwahori--Hecke algebra corresponding to any Coxeter group.
  Deodhar's {\em defect} statistic [{\em Geom.\;Dedicata} {\bf 36}, no.\,1 (1990)]
  %pp.~95--119] 
  allows one to expand products of 
  %modified signless 
  simple Kazhdan--Lusztig basis elements of $H$ in the natural basis of $H$.
  Billey and Warrington [{\em J. Algebraic Combin.} {\bf 13}, no.\,2 (2001)] 
  %pp.~111--136]
  provided a graphical interpretation of the type-$\msfA$ case of this formula.
  Clearwater and the third author
  [{\em Ann.\;Comb.}\;{\bf 25}, no.\,3 (2021)]
  % pp.~757--787]
  %gave a type-$\msfA$ exension
  extended
  the graphical type-$\msfA$ case 
  of this formula 
  %for the special case of $H$ having 
  %in type $\msfA$ to 
%  which 
  to combinatorially expand
  %describe 
  %the natural expansion of 
  products of 
  %more general 
  Kazhdan--Lusztig basis elements 
  indexed by ``smooth"
  %maximal parabolic 
  elements of the symmetric group.
  %in the natural basis of $H$.
  We similarly extend the 
  %give a 
  type-$\msfBC$ case of
  %extension of 
  Deodhar's result to combinatorially expand
  %describe 
  %the natural expansion of 
  products of Kazhdan--Lusztig basis elements indexed by hyperoctahedral group elements which are ``simultaneously smooth" in types $\msfBB$ and $\msfC$.}

%% put your French abstract here, or comment this out if you don't have one
% \resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Hecke algebra, Kazhdan--Lusztig basis, hyperoctahedral group}

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\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{my.bib}
\begin{document}
\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}\label{sec: introduction}
Let $(W,S)$ be a Coxeter system with generating set $S = \{ s_1, \dotsc, s_m \}$, let $\leq$ denote the Bruhat order on $W$, let $H$ be the Iwahori--Hecke algebra corresponding to $W$, and let $\{ T_w \,|\, w \in W \}$ be the {\em natural basis} of $H$ as a $\smash{\zqq}$-module. (See, e.g., \cite{BBCoxeter}.)
A second basis $\{ C'_w(q) \,|\, w \in W \}$ introduced by Kazhdan and Lusztig~\cite{KLRepCH}, sometimes rescaled as 
$\{ \wtc wq \,|\, w \in W \}$ \ntnsp with 
$\wtc wq \defeq q^{\ell(w)/2} C'_w(q)$, 
%${\wtc wq \defeq \qp{\ell(w)}2 C'_w(q)}$, 
%has been the subject of much research because of its importance 
is important in many areas of mathematics.
%the ares of quantum groups, representation theory, and geometry.
%In particular, {\em products} of elements of this
%We will focus on products of the 
%{\em Kazhdan--Lusztig basis} 
%elements, which
Products of these elements expand 
%roducts of these basis elements and their expansions 
nonnegatively in the natural and Kazhdan--Lusztig bases,
and 
%the resulting coefficients
have appeared in intersection homology~\cite{BBDFaisceaux}, \cite{SpringerQACI}, 
%algorithmic and 
combinatorial description of Kazhdan--Lusztig basis elements themselves~\cite{BWHex,Deodhar90}, Schubert varieties~\cite{BWHex}, total nonnegativity~\cite{GJImm,SkanNNDCB,StemImm,StemConj}, 
trace evaluations~\cite{CHSSkanEKL,CSkanTNNChar,GreeneImm,KLSBasesQMBIndSgn,SkanHyperGC}, and chromatic symmetric functions~\cite{CHSSkanEKL,SkanHyperGC}.

We will focus on Deodhar's result
%work in \cite{Deodhar90}.
%($\star$ Motivate the study of products of Kazhdan--Lusztig basis elements with examples from the literature.)
%In 
\cite[Proposition~3.5]{Deodhar90}, where he
considered sequences
%\begin{equation}\label{eq:genseq}
%\mathbf{s} = 
$(s_{i_1},\dotsc,s_{i_k})$
%\end{equation}
of generators in $S$, 
products of the corresponding 
Kazhdan--Lusztig basis elements $\wtc{s_{i_j}\ntksp}q = T_e + T_{s_{i_j}}$ of 
%a
%the Kazhdan--Lusztig basis of 
$H$,
%certain modified signless {\em Kazhdan--Lusztig} basis $\{ \wtc wq \,|\, w \in W \}$ of $H$~\cite{KLRepCH},
and their natural expansions
%of the 
%corresponding 
%resulting products
\begin{equation}\label{eq:Deoprod}
\wtc{s_{i_1}\ntksp}q \cdots \wtc{s_{i_k}\ntksp}q
%    (T_e + T_{s_{i_1}}) \cdots (T_e + T_{s_{i_k}}) 
= \sum_{w \in W} a_w T_w.
\end{equation}
%in $H$.  
%Each factor on the left-hand side of (\ref{eq:Deoprod}) is a multiple of an element of the signless {\em Kazhdan--Lusztig basis}~\cite{KLRepCH} of $\hnq$. 
% Deodhar described the coefficients $\{ a_w \,|\, w \in W \} \subset \mathbb Z[q]$ 
% as follows.
% Define a {\em subexpression} 
% of $(s_{i_1}, \dotsc, s_{i_k})$
% to be a sequence $\sigma = (\sigma_0, \sigma_1, \dotsc, \sigma_k)$ of elements of $W$ satisfying $\sigma_0 = e$ and
%  $\sigma_j \in \{ \sigma_{j-1}, \sigma_{j-1}s_{i_j}\}$ for $j = 1,\dotsc, k$.
% Call index $j$ a {\em defect} of $\sigma$ if $\sigma_{j-1}s_{i_j} < \sigma_{j-1}$ 
% and let $\dfct(\sigma)$ denote the number of defects of $\sigma$. Then each coefficient on the right-hand side of (\ref{eq:Deoprod}) is given by
% \begin{equation}\label{eq:Deodefectformula}
%     a_w = \sum_\sigma q^{\dfct(\sigma)},
% \end{equation}
% where the sum is over all subexpressions
% $\sigma$ of $(s_{i_1},\dotsc, s_{i_k})$
% satisfying $\sigma_k = w$.
Deodhar described the coefficients $\{ a_w \,|\, w \in W \} \subset \mathbb Z[q]$ in terms of
%as follows.
%Define a {
{\em subexpressions} 
of $(s_{i_1}, \dotsc, s_{i_k})$:
sequences $\sigma = (\sigma_1, \dotsc, \sigma_k)$ of elements of $S$ with
%$\sigma_0 = e$ and
 $\sigma_j \in \{ e, s_{i_j}\}$ for $j = 1,\dotsc, k$.
Call index $j$ a {\em defect} of $\sigma$ if $\sigma_1 \cdots \sigma_{j-1}s_{i_j} < \sigma_1 \cdots \sigma_{j-1}$,
and let $\dfct(\sigma)$ denote the number of defects of $\sigma$. 
%(Our notation here differs from that in \cite{Deodhar90}.)
Then each coefficient on the right-hand side of (\ref{eq:Deoprod}) is given by
\begin{equation}\label{eq:Deodefectformula}
    a_w = \sum_\sigma q^{\dfct(\sigma)},
\end{equation}
where the sum is over all subexpressions
$\sigma$ of $(s_{i_1},\dotsc, s_{i_k})$
satisfying $\sigma_1 \cdots \sigma_k = w$.
%($\star$ Maybe use a symbol other than $\sigma$ above, in case we want to later use $\sigma$ as a path family of type $s$ for some generator $s$.  Or will these be so far apart that it doesn't matter?)

Billey and Warrington~\cite[Remark~6]{BWHex}
observed that when $W$ and $H$ are
the symmetric group $\sn$ and type-$\msfA$ Iwahori--Hecke algebra
$\hanq$, Deodhar's defect statistic has the following simple graphical interpretation.
Let $\smash{F = F_{s_{i_1}} \circ \cdots \circ F_{s_{i_k}}}$ be the wiring diagram corresponding to $(s_{i_1}, \dotsc, s_{i_k})$, where $\circ$ denotes concatenation, and the factor wiring diagrams 
$F_{s_1}, \dotsc, F_{s_{n-1}}$
are the planar networks
%Associate (type-$\msfA$) wiring diagrams to generators of $\sn$ by 
\begin{equation}\label{eq:A generator wirings}
%F_{s_1}=
\begin{tikzpicture}[scale=.5,baseline=5]%s1
  \node at (-1.4,3.5) {$\scriptstyle n$};
  \node at (1.4,3.5) {$\scriptstyle n$};
  \node at (-1.5,2.5) {$\scriptstyle n-1$};
  \node at (1.5,2.5) {$\scriptstyle n-1$};
  \node at (0,1.75) {$\vdots$};
  %\node at (0,1.1) {$\scriptstyle .$};
 % \node at (0,0.85) {$\scriptstyle .$};
  \node at (-1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-1.5) {$\scriptstyle 1$};
  \node at (1.4,-1.5) {$\scriptstyle 1$};
   \draw[-, very thick]
  (-.5,3.5) -- (.5,3.5);
  \draw[-, very thick]
  (-.5,2.5) -- (.5,2.5);
   \draw[-, very thick]
  (-.5,0.5) -- (.5,0.5);
  \draw[-, very thick]
  % (-1,-0.5) -- (-.5,-1) -- (0,-.5) -- (.5,-1) -- (1,-0.5);
    (-.5,-0.5) -- (0,-1) -- (0.5,-.5);
  \draw[-, thick]
  (-.5,-1.5) -- (0,-1) -- (0.5,-1.5);
\end{tikzpicture}
\ntksp, \quad 
%F_{s_2}=
\begin{tikzpicture}[scale=.5,baseline=5]%s2
  \node at (-1.4,3.5) {$\scriptstyle n$};
  \node at (1.4,3.5) {$\scriptstyle n$};
  \node at (-1.5,2.5) {$\scriptstyle n-1$};
  \node at (1.5,2.5) {$\scriptstyle n-1$};
  \node at (0,1.75) {$\vdots$};
%  \node at (-1.0,1.7) {$\scriptstyle n$};
%  \node at (1.0,1.7) {$\scriptstyle n$};
%  \node at (0,1.35) {$\scriptstyle .$};
%  \node at (0,1.1) {$\scriptstyle .$};
%  \node at (0,0.85) {$\scriptstyle .$};
  \node at (-1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-1.5) {$\scriptstyle 1$};
  \node at (1.4,-1.5) {$\scriptstyle 1$};
     \draw[-, thick]
  (-.5,3.5) -- (.5,3.5);
  \draw[-, very thick]
  (-.5,2.5) -- (.5,2.5);
%   \draw[-, thick]
%  (-.5,1.7) -- (.5,1.7);
   \draw[-, very thick]
  (-.5,0.5) -- (0,0) -- (.5,0.5);
  \draw[-, very thick]
    (-.5,-0.5) -- (0,0) -- (0.5,-.5);
  \draw[-, very thick]
  (-.5,-1.5) -- (0.5,-1.5);
\end{tikzpicture}\ntksp, 
%\\
 \dotsc, 
 %F_{s_{n-1}}=
\begin{tikzpicture}[scale=.5,baseline=5]%sn-1
  \node at (-1.4,3.5) {$\scriptstyle n$};
  \node at (1.4,3.5) {$\scriptstyle n$};
  \node at (-1.5,2.5) {$\scriptstyle n-1$};
  \node at (1.5,2.5) {$\scriptstyle n-1$};
    \node at (0,1.75) {$\vdots$};
%  \node at (0,0.35) {$\scriptstyle .$};
%  \node at (0,0.1) {$\scriptstyle .$};
%  \node at (0,-0.15) {$\scriptstyle .$};
    \node at (-1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,0.5) {$\scriptstyle 3$};
  \node at (1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-0.5) {$\scriptstyle 2$};
  \node at (-1.4,-1.5) {$\scriptstyle 1$};
  \node at (1.4,-1.5) {$\scriptstyle 1$};
   \draw[-, very thick]
  (-.5,3.5) -- (0,3) -- (.5,3.5);
   \draw[-, very thick]
  (-.5,2.5) -- (0,3) -- (.5,2.5);
   \draw[-, very thick]
  (-.5,0.5) -- (.5,0.5);
  \draw[-, very thick]
    (-.5,-0.5) -- (0.5,-.5);
  \draw[-, very thick]
  (-.5,-1.5) -- (0.5,-1.5);
\end{tikzpicture}\ntksp,
\end{equation}
respectively.  Edges of $F$ are understood to be oriented from left to right, with $n$ {\em source}
vertices on the left,
%called {\em sources}, and 
$n$ {\em sink} vertices on the right,
%called {\em sinks}. 
and $k$ more {\em interior} vertices, one per factor. %$F_{s_{i_j}}$.  
%$F$ can be covered by 
We can cover $F$ with $2^k$ different path families of the form $\pi = (\pi_1,\dotsc,\pi_n)$, if we allow 
%the $n$ component paths
%start at sources $1, \dotsc, n$, respectively,
%and 
two paths meeting at the 
%central
interior vertex of a factor $\smash{F_{s_{i_j}}}$ either to cross or not to cross there. 
%If the paths terminate at sinks $w_1, \dotsc, w_n$, respectively, for some $w \in \sn$,
%then we say that $\pi$ has {\em type $w$}.
Let $\Pi_w(F)$ be the subset of these path families 
having {\em type $w$}, i.e., for $i = 1,\dotsc, n$, path $\pi_i$ begins at source $i$ 
%of $F$ 
and terminates at sink $w_i$. 
Call index $j$ a {\em (type-$\msfA$) defect} of $\pi$ if the two paths meeting %at the central vertex of 
in $\smash{F_{s_{i_j}}}$ 
%have crossed 
cross an odd number of times
in 
%the subdiagram 
$\smash{F_{s_{i_1}} \circ \cdots \circ F_{s_{i_{j-1}}}}$, 
%to the left of $F_{s_{i_j}}$, 
and let $\dfct(\pi)$ be the number
of defects in $\pi$. Then each coefficient on the right-hand side of (\ref{eq:Deoprod}) is given by
\begin{equation}\label{eq:BWdefectformula}
%      a_w = \sum_\pi q^{\dfct(\pi)},
    a_w = \sum_{\pi\in \Pi_w(F)} 
q^{\dfct(\pi)}.
\end{equation}
%where the sum is over all path families $\pi$ covering $F$ and having {\em type $w$}, i.e., for $i = 1,\dotsc, n$, path $\PiBC_i$ begins at source $i$ 
%of $F$ and terminates at sink $w_i$. 
%($\star$ Could define $\PiBC_w(F)$ in this section, and write $\pi \in \PiBC_w(F)$ in the summation)
%of $F$ 
%for $i = 1,\dotsc, n$.
%and having type $w$.
%$\sigma$ of 
%$\mathbf{s}$
%$(s_{i_1} \cdots s_{i_k})$
%satisfying $\sigma_k = w$.

Clearwater and the third author~~\cite[Corollary~5.3]{CSkanTNNChar} extended this type-$\msfA$ result to products of the form
\begin{equation}\label{eq:CSprod}
\wtc{v^{(1)}}q \cdots \wtc{v^{(k)}}q
%\bigg( \sum_{u \leq v^{(1)}} T_u \bigg) 
%\cdots
%\bigg( \sum_{u \leq v^{(k)}} T_u \bigg)
= \sum_w a_w T_w 
\end{equation}
in $\hanq$,
where $v^{(1)}, \dotsc, v^{(k)}$
are maximal elements of parabolic subgroups of $\sn$ (and more generally, \pavoiding elements of $\sn$), and 
each factor
\begin{equation*}
    \wtc{v^{(j)}}q \defeq \sum_{u \leq v^{(j)}} \ntksp T_u
\end{equation*}
%are more general elements of the modified signless Kazhdan--Lusztig basis of $\hnq$
%~\cite[Corollary~5.3]{CSkanTNNChar}.
%Again, these factors are multiples of the signless Kazhdan--Lusztig
%basis of $\hnq$.
%the corresponding 
belongs to the rescaled
%modified signless 
Kazhdan--Lusztig basis of $\hanq$.
Again we have (\ref{eq:BWdefectformula}), where 
$F$ has the form
%the sum
%is over path families of type $w$ which cover a certain planar network of the form
$F_{v^{(1)}} \circ \cdots \circ F_{v^{(k)}}$, 
with factor networks
%in which 
%and the
%factors $F_{v^{(j)}}$ 
%elements satisfy
%which 
%belonging to a class of wiring diagrams
%has a form generalizes
%The planar networks used generalize 
%the wiring diagrams in
%constructed from 
generalizing those in
(\ref{eq:A generator wirings}) by allowing interior vertices to have indegree and outdegree greater than $2$.
%more than two paths to cross at a vertex.

We extend this result further to analogous products of elements
of the Kazhdan--Lusztig basis of the type-$\msfBC$ Iwahori--Hecke algebra 
$\hbnq$
in \Cref{s:main}.
To do so, we review
%we review
necessary background material on 
the hyperoctahedral group,
$\hbnq$, 
and type-$\msfBC$ planar networks
%and $\hbnq$
%the type-$\msfBC$ Hecke algebra
in Sections~\ref{s:bn} -- \ref{s:BCstarnets}.
%Then we present our main results in \Cref{s:main}.


\section{The hyperoctahedral group}
%and its Iwahori--Hecke algebra}
\label{s:bn}
%expression $\mathbf{s}$
Recall that for a Coxeter group $W$ with generating set $S$
and an element $w \in W$, an expression $w = s_{i_1} \cdots s_{i_\ell}$ for $w$ as a product of generators is called {\em reduced}
if it is as short as possible, and $\ell = \ell(w)$ is called the {\em length} of $w$.  We define the {\em Bruhat order} on $W$ by
declaring $v \leq w$ if some (equivalently, every) reduced expression
for $w$ contains a subsequence which is a reduced expression for $v$.

For a $J \subseteq S$
%an arbitrary Coxeter group $W$ and 
%subset $J$ of $S$,
let $\smash{W_J}$ denote the 
{\em parabolic} 
subgroup of $W$ generated by $J$.
Each coset $w\smash{W_J}$ of $W_J$ has a unique Bruhat-minimal element \cite[Corollary~2.4.5]{BBCoxeter}. 
% a unique Bruhat-maximal element \cite[Corollary~2.4.5]{BBCoxeter}.
Let $\smash{\Wejm}$ denote the set of minimal coset representatives.
It is known that we have \cite[Definition~2.4.2, Lemma.~2.4.3]{BBCoxeter} 
%that we have
%to be the set of minimal-length representatives of cosets of the form $wW_J$,
\begin{equation}\label{eq:wejmdefns}
%\begin{aligned}
    \Wejm = \{ w \in W \,|\, ws > w \text{ for all } s \in J \} 
    = \{ w \in W \,|\, wv > v \text{ for all } v \in W_J \} 
%    &= \{ w \in W \,|\, \text{$i$ appears before $i+1$ in $w$
%    \end{aligned}
\end{equation}
and that
%For each subset $J$ of generators, 
there is a bijection~\cite[Proposition~2.4.4]{BBCoxeter}
\begin{equation}\label{eq:Jfactor}
%\begin{aligned}
    \Wejm \times W_J \;
    \overset{1-1}{\longleftrightarrow} \;
    W
    %\bn
    \end{equation}
    given by simple multiplication $(w,u) \mapsto wu$.
The Bruhat order on $W$ induces a related partial order on the set $W / W_J$ of cosets $\{ wW_J \,|\, w \in \bn \}$: we declare $vW_J \leq wW_J$ if the minimal element of $vW_J$ is less than or equal to the minimal element of $wW_J$.  (Equivalently, we may define $\leq$ by comparing 
% maximal elements or 
arbitrary elements of the two cosets \cite[Lemma~2.2]{DougInv}.)
We call this poset the {\em Bruhat order on $W / W_J$}.
%Let $W$ be a Coxeter group with generating set $S$
%For any 
%arbirary 
%Coxeter group $W$ and subset $J$ of its generators,

The {\em hyperoctahedral group}
$\,\bn$ is the Coxeter group of type $\mathsf B_n = \mathsf C_n$,
%i.e., the 
%$\bn$ is the centralizer in $\snn$
%\smash{\defeq} \mfs{[-n, -1] \cup [1,n]}$
%of the element $(1,-1)(2,-2)\cdots (n, -n)$.
%We may view $\bn$ as the subgroup of $\snn$
%It is 
generated by $S=\{s_0,\dotsc,s_{n-1}\}$, subject to relations
%\begin{equation*}
%  t\ (= s'_0) \defeq s_0, \qquad
%  s'_i \defeq s_is_{\smash{\ol i}}, \text{ for } i = 1,\dotsc,n-1,
%\end{equation*}
%which satisfy the relations
%$s_0$ and $s_is_{\smash{\ol i}}$, $i = 1, \dotsc, n-1$ of $\snn$.
%We will find it convenient to define
%Abstractly, $\bn$ is generated by these elements,
%%$t=s_0, s_1, \dotsc, s_{n-1}$,
%subject to the relations
\begin{equation}\label{eq:bnpresentation}
  \begin{alignedat}2
    {s_i}^2 &= e &\quad &\text{for $i = 0, \dotsc, n-1$,}\\
    s_0s_1s_0s_1 &= s_1s_0s_1s_0, &\quad & \\
%    ts_i &= s_it &\quad &\text{for $i=2,\dotsc,n-1$,}\\
    s_is_j &= s_js_i &\quad &\text{for $i,j \geq 0$ and $|i-j| \geq 2$,}\\
    s_is_js_i &= s_js_is_j &\quad &\text{for $i,j \geq 1$ and $|i-j| = 1$.}
  \end{alignedat}
\end{equation}
%If an expression $s_{i_1} \cdots s_{i_\ell}$ for $w \in \bn$
%is as short as possible,
%then we call it {\em reduced} and we call $\ell = \ell(w)$ the {\em length} of $w$.
%More generally, a Coxeter group is often denoted by $W$ with a generating set $S=\{s_0,s_1, \ldots, s_{n-1}\}$. Let $J\subseteq S$ be a subset of generators, 
%$W_J$ denotes 
%the subgroup of $W$ generated by $J$ is called {\em parabolic} and is denoted by $W_J$.
%Thus one can view the symmetric group $\sn$, the Coxeter group of type $\msfA_{n-1}$, as the parabolic subgroup $W_J$ of $W = \bn$ generated by $J = \{ s_1,\dotsc,s_{n-1} \}$.
%often uses
The parabolic subgroup of $\bn$ generated by $\{s_1,\dotsc,s_{n-1}\}$ is the {\em symmetric group} $\sn$, the Coxeter group of type $\msfA_{n-1}$.

Like elements of 
$\sn$,
%the symmetric group $\sn$, the Coxeter group of type $\msfA_{n-1}$,
elements of 
%the hyperoctahedral group 
$\bn$
naturally correspond to permutations of
letters belonging to a certain alphabet.
%acts on a certain set of permutations.
%elements may be viewed as certain permutations.
%To be precise, 
Specifically, we define the alphabet
\begin{equation*}
  [\ol n, n] \defeq \{ -n, \dotsc, n \} \ssm \{ 0 \}
  \end{equation*}
%of letters, 
with notation $\ol a \defeq -a$ for all $a \in [\ol n,n]$,
and we consider permutations
%each and for each letter $a \neq 0$ we let
%For $a$ in this set, we will find it convenient to define
%$\ol a$ denote $-a$.
%for all $a \in [\ol n,n]$.
%We associate elements of $\bn$ to permutations
$w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n$ of $[\ol n, n]$
%of the word
%$\ol n \cdots \ol 1 1 \cdots n$
which satisfy $w_{\ol i} = \ol{w_i}$ for all $i \in [\ol n, n]$.
We define a correspondence between $\bn$ and such permutations via the (left) action of $\bn$ on these permutations:
%defined by
%    $(1)$ 
$s_0$ swaps the letters in positions $\ol1$ and $1$; 
%and   
%    $(2)$ 
$s_i$ ($i = 1,\dotsc,n-1$) simultaneously swaps 
    the letters in positions $i$, $i+1$ with each other,
    and
    the letters in positions $\ol i$, $\ol{i+1}$ with each other.
Then for $w = s_{i_1} \cdots s_{i_r} \in \bn$,
we define the {\em (long) one-line notation} of $w$ to be the permutation
%letting $s_i$, $t$ act on the
%\begin{equation*}
%  \begin{gathered}
%    s_i(\ol n \cdots \ol 1 1 \cdots n) =
%    \ol n \cdots \ol{(i+2)} \ol i \ol {(i+1)} \ol{i-1} \cdots \ol 1
%    1 \cdots (i-1) (i+1) i (i+2) \cdots n,\\
%    t(\ol n \cdots \ol 1 1 \cdots n) =
%    \ol n \cdots \ol 2 1 \ol 1 2 \cdots n,
%  \end{gathered}
%\end{equation*}
%and by defining the one-line notation of $s_{i_1} \cdots s_{i_r}$ to be
\begin{equation}\label{eq:s2nrearrange}
  w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n 
= s_{i_1}(s_{i_2}( \cdots (s_{i_r}( \ol n \cdots \ol 1 1 \cdots n)) \cdots )).
\end{equation}
%$\star$ Do an example or two.
For example, when $n=4$, the element $s_0s_1 \in \mfb4$
%\in \mfs8$
%= \mfs{2\cdot4}$
has long one-line notation
\begin{equation}\label{eq:long1lineex}
%  \begin{aligned}
    s_0( s_1(\ol4 \ol3 \ol2 \ol1 1234))
    = s_0(\ol4 \ol3 \ol1 \ol2 2 1 3 4)
    = \ol4 \ol3 \ol1 2 \ol2 1 3 4.
%    = \ol4\ol3 2 \ol2 \ol1 1 3 4.
%  \end{aligned}
  \end{equation}
%($\star$ Remove this comment and just use intervals always?)
%To resolve the ambiguity caused by our notation $\snn$, we will
%refer to index sets when necessary, e.g., rather than writing $\mfs4$,
%we will write
%$\mfs{[\ol2, 2]}$ for the group of permutations of $\{\ol2, \ol1, 1, 2\}$,
%and $\mfs{[1,4]}$ for the group of permutations of $\{1, 2, 3, 4\}$.
%instead of $\mfs{4}$.
% (By our definition, the right action of $s_0$ swaps the letters $1, \ol 1$,
% wherever they are, while the right action of $s_i$ for $i = 1,\dotsc,n-1$ swaps
% the letters $i$, $i+1$, wherever they are, while simultaneously swapping the letters $\ol i$, $\ol{i+1}$,
% wherever they are.)
%For integers $p, r \in \mathbb Z \ssm \{0\}$, we has odd length
It follows that $w_i^{-1}$ is the index $j$ satisfying $w_j = i$.
%We can also compute $\ell(w)$ by 
By %examining
counting certain 
inversions in the long one-line notation of $w$, we can compute $\ell(w)$.
\begin{lem}\label{l:longinversions}
The length of $w \in \bn$ equals the number of pairs $(i,j)$ with
%\begin{enumerate}
    %\item 
    $|i| \leq j$ and $j$ appearing earlier than $i$ in
    $w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n$.
    \end{lem}
    \begin{proof}
% %    $\star$ Prove this.
% Let $\ell'$ be the proposed formula for length,
% %number of pairs $(i,j)$ described above, 
% and assume that $\ell'$ agrees with the true length $\ell$ for
% %Assume that the claim is true for 
% hyperoctahedral groups $\mathfrak B_1,\dotsc, \mathfrak B_{n-1}$.
% Consider 
% %$w \in \bn$.  
% %Suppose that $n$ appears in position $k$
% %of the two-line notation of $w$
% $w  = w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n \in \bn$ satisfying
% %with letter $n$ appearing in some position
% $w_k = n$, $w_{\ol k} = \ol n$ for some 
% $k \in [\ol n,n]$, and define $v = v_{\ol{n-1}} \cdots v_{\ol1} v_1 \cdots v_{n-1} \in \mathfrak B_{n-1}$ 
% by removing letters $n$, $\ol n$ from $w$.
% %be the element obtained from $w$
% %from this long one-line notation 
% %is obtained from that of $w$ 
% %by removing the letters $n$ and $\ol n$.  
% By \Cref{l:BClength} we have 
% %that $\ell(w) - \ell(v)$ is the number of inversions
% %in $w_1 \cdots w_n$ involving $n$ or $\ol n$, plus %$n$ if $\ol n$ appears here, i.e.,
% \begin{equation}\label{eq:lengthinduction}
% %\begin{aligned}
%     \ell(w) = \begin{cases}
% %    \ell(v) + \# \{ i \,|\, k < i \leq n \}
% %    &\text{if $k > 0$,}\\
% %    \ell(v) + \# \{ i \,|\, 1 \leq i < k \} + n
% %    &\text{if $k < 0$.}
% %    \end{cases}\\
% %    &= \begin{cases}
%         \ell(v) + n-k &\text{if $k >0$},\\
%        \ell(v) + |k|-1 + n &\text{if $k < 0$},
%        \end{cases}
% %       \end{aligned}
% \end{equation}
% since $n$ forms inversions with $w_{k+1}, \dotsc, w_n$ %precedes $n-k$ smaller letters in $w_1 \cdots w_n$ 
% if it appears in position $k > 0$, and 
% $\ol n$ forms inversions with $w_1 \cdots w_{|k|-1}$ if it appears in position $|k|$ with $k < 0$.
% %$|k|-1$ letters precede $\ol n$ if $n$ appears in  
% %position $k < 0$.  
% On the other hand, by the definition of $\ell'$ and induction we have
% \begin{equation*}
% \begin{aligned}
%        \ell'(w) &= 
%        \ell'(v) + \# \text{ letters following $n$ in $w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n$} \\
%        &= \ell(v) + \# \text{ letters following $n$ in $w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n$},
% %       \begin{cases}
% %        \ell'(v) + n-k &\text{if $k > 0$},\\
% %        \ell'(v) + k-1 + n &\text{if $k < 0$}.
% %    \end{cases}
%     \end{aligned}
% \end{equation*}
% which equals the right-hand side of (\ref{eq:lengthinduction}).
% %Since $\ell(v) = \ell'(v)$ for $v \in \mathfrak B_{n-1}$, we have the desired result.
% %whose
% %Remove the letters $\ol n$, $n$ from the one-line notation of $w$ to obtain $v \in \mathfrak B_{n-1}$...
%     %    (When $i>0$, $(i,j)$ is an inversion, contributing to the first term in $\ell(w)$. This must mean that when $i<0$, $(i,j)$ contributes to the second term in $\ell(w)$.)
% %    Maybe useful fact: If we assume $|i| \neq |j|$, then when we count some pair ($i,j$) with $|i| \leq j$ but $w_i > w_j$, we can also count a pair ($\ol i, \ol j$) with $w_{\ol j} > w_{\ol i}$.
Omitted.
    \end{proof}
%($\star$ Motivate and introduce the terminology "short" and "long" one-line notation.  Use the three permutations in (\ref{eq:long1lineex}) as a brief example. - Done, please review)
The condition $w_{\ol i} = \ol{w_i}$
implies that each element (\ref{eq:s2nrearrange}) is  
%We can see that each of these permutation is 
completely determined by the subword $w_1 \cdots w_n$, called the
%letters in positions $1,\dotsc, n$, since $w_{\ol i} = \ol{w_i}$ for all $i \in [\ol n, n]$. We let the $2n$-letter permutation defined in (\ref{eq:long1lineex}) be the \textit{long} one-line notation of $w$. For brevity, we define the 
\textit{short one-line notation} of $w$.
%to be the subpermutation
%\begin{equation}\label{eq:snrearrange}
%  w_1 \cdots w_n 
%= s_{i_1}(s_{i_2}( \cdots (s_{i_r}( 1 \cdots n)) \cdots )),
%\end{equation}
%consisting of the letter in indices $1,\dotsc, n$ of the long one-line %notation.
%Whether a given permutation is represented in short vs. long one-line notation is clear from context, such as the value of $n$.
The set of short one-line notations of elements of $\bn$ is the set of {\em signed permutations} of $[1,n]$:
words $w_1 \cdots w_n$ with letters in the alphabet $[\ol n,n]$ 
with no repeated absolute values.
%such that the absolute values $\{|w_1|, \dotsc, |w_n| \}$ are precisely $[1,n]$.
%The correspondence between $\bn$ and signed permutations is given by the left action of $\bn$ on signed permutations defined by
%
%$(1)$ $s_0$ negates the letter in %position $1$, and
%  
%    $(2)$ $s_i$ ($i = 1,\dotsc,n-1$) swaps the letters in positions $i$, $i+1$ with each other.
%
%For example the short one-line %notation $\ol2134$ of the element 
%$s_0s_1 \in \mfb{4}$
%(\ref{eq:long1lineex}) 
%in $\mfb4$ has short one-line notation 
%can be obtained by its action on %$1234$: 
%the action
%\begin{equation}\label{eq:short1lineex}
%   s_0( s_1(1234))
%    = s_0(2 1 3 4)
%    = 
%\ol2 1 3 4.
%\end{equation}
%(By our definition, the right action of $s_0$ on signed permutations negates the letter $1$ (or $\ol1$), wherever it is,
%while the right action of $s_i$ for $i = 1,\dotsc,n-1$ swaps the letters having absolute values
%$i$, $i+1$, wherever they are, while leaving any negation signs in their original positions.)  

%We define an \emph{inversion} of any word $w_1 \cdots w_n$
%to be a pair $(i,j)$ of indices 
%satisfying $i < j$ and $w_i > w_j$.  
%Let $\inv(w)$ denote the number of inversions in $w$.
%For $w \in \sn$, it is well known that we have $\inv(w_1 \cdots w_n) = \ell(w)$.
%For $w \in \bn$ we can similarly compute $\ell(w)$ by inspecting the short one-line notation of $w$.  (See, e.g., \cite[Proposition~8.1.1,  Equation~(8.2)]{BBCoxeter}.)
%\begin{lem}\label{l:BClength}
%  Let $w \in \bn$ have short one-line notation $w_1 \cdots w_n$.
%  Then its length satisfies
%  \begin{equation*}
%    \ell(w) = \inv(w_1 \cdots w_n) + \sum_{i: w_i < 0} |w_i|.
%  \end{equation*}
%\end{lem}

%\end{enumerate}
%($\star$ Define subintervals $[a,b]$ of $[\ol n, n]$.  Define reversals of the forms $s_{[\ol a, a]}$ for $a = 1,\dotsc, n$
%and $s_{[a,b]}$ for $1 \leq a < b \leq n$ both in terms of one-line notation and in terms of generator notation. DONE. Not checked. GH: Should we use short 1-line notation instead?
%Should this come before or after the lemma?
%)

%($\star$ How about using the sentences at the beginning of Section 2 of 2020bc.tex to define intervals?)

Certain elements of the hyperoctahedral group are most easily defined in terms of subintervals of $[\ol n, n]$, where we declare any subset 
%\begin{equation*}
    $[a,b] \defeq \{ a, \dotsc, b \} \ssm \{0 \}$
%\end{equation*}
 of $[\ol n,n]$ 
 to be 
 an {\em interval}, even if $a < 0 < b$.
% \begin{equation*}
%   [a, b] 
%   \defeq \left\{ \begin{array}{cc}
%       \{a, \dotsc, b\}  & \text{ if } a,b<0 \text{ or } 0<a,b  \\
%        \{a, \dotsc, \ol 1, 1, \dotsc, b \} & \text{ if } a<0<b
%   \end{array}\right. (\star Review? - See\ comment)
%   % \{ a, \dotsc, b \} \ssm \{ 0 \}
%   \end{equation*}
% to be a subinterval of $[\ol n, n]$ for any nonzero $a,b$ such that $\ol n \leq a < b \leq n$. 
Roughly speaking, we define a {\em reversal} of $\bn$ to be an element $s_{[a,b]}$ obtained from the identity by reversing letters $[a,b]$ in positions $[a,b]$, and ensuring that the resulting permutation belongs to $\bn$.
To be precise, we describe such elements using
% \emph{type-$\msfBC$ satisfactory} intervals. An interval $[a,b]$ is type-$\msfBC$ satisfactory under
 three cases: $a=b$, $a = \ol b$ ($b > 0$), and $0 < a < b$.
 % ($\star$ Should we mention the trivial case of $s_{[a,b]} = e$? Something like: Note that if $a=b$, we have $s_{[a,b]} = s_\emptyset = e$, which trivially is a reversal.)
%For $a = \ol b$ %TODO - check this phrasing. I think writing the case as $a = \ol b$ is more clear
When $a=b$, we have the trivial reversal $s_{[b,b]} = s_\emptyset = e$. %which trivially is a reversal. 
For $b>0$, the reversal $s_{[\ol b,b]}$ is the element having 
%short 
one-line notation
\begin{equation}
%    \ol 1 \cdots \ol b \cdot (b+1) \cdots n.
 \ol n \cdots \ol{(b+1)} \cdot b \cdots 1 \cdot \ol 1 \cdots \ol b \cdot (b+1) \cdots n,
\end{equation}
and equal to the product 
%In terms 
of generators
%, we have
%\begin{equation}
%    s_{[\ol b,b]} = 
$s_0 (s_1 s_0 s_1)(s_2s_1s_0s_1s_2) \cdots (s_{b-1} \dotsc s_1 s_0 s_1 \dotsc s_{b-1})$.
%\end{equation}
%of the form $s_{[\ol b, b]}$ ($b > 0$) or $s_{[a,b]}$ ($0 < a < b$) 
%containing
%whose (long) one-line notation contains letters $[a,b]$
%consecutively and in reverse order, while each other letter $i$ appears in %position $i$.
%of the form
%notation
%We further define the reversals
%\begin{equation*}
%\begin{aligned}
%    s_{[\ol a, a]} & := \{\ol n, \ol{n-1}, \dotsc, \ol{a+1}, a, a-1, \dotsc , 1, \ol 1, \ol 2, \dotsc \ol a, a+1, \dotsc, n\} \\
%                & = 
%            \end{aligned}
%\end{equation*}
%and
For $0 < a < b$ the reversal $s_{[a,b]}$ is the element having 
%short 
one-line notation
\begin{equation*}
%1 \cdots (a-1) \cdot b \cdots a \cdot (b+1) \cdots n.
\ol n \cdots \ol{(b+1)} \cdot \ol a \cdots \ol b \cdot \ol{(a-1)} \cdots \ol 1
\cdot 
1 \cdots (a-1) \cdot b \cdots a \cdot (b+1) \cdots n,
\end{equation*}
and equal to the product
%In terms 
of generators
%, we have
%\begin{equation*}
%    \begin{aligned}
%    s_{[a,b]}  =
    %\\
    %&:={\small \{\ol n, \dotsc, \ol{b+1}, \ol a, \ol{a+1}, \dotsc \ol{b-1}, \ol b, \ol{a-1}, \dotsc, \ol 1, 1, \dotsc, a-1, b, b-1, \dotsc, a+1, a, b+1, \dotsc n\}} \\
     %&= 
     $s_a (s_{a+1}s_a)(s_{a+2}s_{a+1}s_a) \cdots ( s_{b-1} \cdots s_a)$. 
%     s_{b-1} (s_{b-2} s_{b-1}) \dotsc (s_{a+1} s_{a+2} \dotsc s_{b-1})(s_{a} s_{a+1} \dotsc s_{b-1})
    %\end{aligned}
%\end{equation*}
% Gavin wrote these, someone else check me please! Thanks! The flipped should also work, and this is the shortest form of several. 
% Should we use short 1-line notation here instead? 
% TJP - I think short 1-line would be good here. Also 1-line should solely be a word, without brackets, as I understand. We might rewrite like:
% The one-line notation of $s_{[a,b]}$ is $1 \cdots (a-1) b \cdots a (b+1) \cdots n$.
% The one-line notation of $s_{[\ol a,a]}$ is $\ol 1 \cdots (\ol{a-1}) \ol a (a+1) \cdots n$.
% We define a \textit{reversal} $s_{[a,b]} \in \bn$ to be an element by which left-multiplication reverses the subpermutation on $[a,b]$ of an element's one-line notation.
% The one-line notation of $s_{[a,b]}$ is $w_{\ol n} \cdots w_{a-1} w_b \cdots w_{a} w_{b+1} \cdots w_n$. % This is incorrect - must revise
%($\star$ Define the Bruhat order on $\bn$ in terms of subexpressions of products of generators.  Is this in 2020bc.tex? Then just mention
%that the Bruhat order can also be defined in terms of one-line notation and refer the reader to Proctor's paper~\cite{ProctorBruhat}.)
%($\star$ Please check/revise definitions of $\mfb{[a,b]}$ and $J_{[a,b]}$.)
%($\star$ Will these definitions be necessary?)
%($\star$ These are referenced in the proof of Proposition~\ref{p:BCbijectiontoshorterpathreversals}.)
Each reversal $s_{[a,b]}$ is the unique element of maximum length in a parabolic subgroup of $\bn$, generated by
\begin{equation}\label{eq:jabdef}
    J_{[a,b]} \defeq \begin{cases}
        \{ s_0, \dotsc, s_{b-1} \} &\text{if $a = \ol b$ ($b > 0$)},\\
        \{ s_a, \dotsc, s_{b-1} \} &\text{if $0 < a < b$}.
        \end{cases}
\end{equation}
% Define $J_{[a,b]} \subseteq S$ to be the subset of generators covering (or acting on?)
% % covering % Probably should use more precise word than 'covering'
% the
% % type-$\msfBC$ satisfactory
% interval $[a,b]\subseteq [\ol{n},n]$.
% % .
% % , when $a = \ol b$ ($b > 0$), or $0 < a < b$.
% If $a = \ol b \,(b>0)$, we have $J_{[a,b]} = \{s_0, s_1, \dotsc, s_{b-1}\}$. 
% Otherwise, i.e. $0 < a < b$, we have $J_{[a,b]} = \{s_a, \dotsc, s_{b-1}\}$. Define $\mfb{[a,b]}: = \mfb{J_{[a,b]}}$ to be the parabolic subgroup of $\mfb{n}$ generated by $J_{[a,b]}$. 
% % Note that every element $w \in \mfb{[a,b]}$ satisfies $w \leq s_{[a,b]}$.
% ($\star$ Use se $W_J$ later instead of $\mathfrak{B}_{[a,b]}$.)
%For any Coxeter group $W$, d
%Define the 
%{\em Bruhat order} on $W = \bn$
%$<$, is the poset defined 
%by declaring $v \leq w$ if some (equivalently, every) reduced expression for $w$ contains a reduced expression for $v$. 
%The Bruhat order can also be defined in terms of signed permutations. (See \cite[Theorem~5BC]{ProctorBruhat} for details.)
%It is known~\cite[Corollary~2.2.5]{BBCoxeter} that the Bruhat order satisfies
%\begin{equation}\label{eq:bruhatinverse}
%    v \leq w \ \Longleftrightarrow\ v^{-1} \leq w^{-1}.
%\end{equation}
%the short one-line notation
%Given a parabolic subgroup $\smash{W_J}$,
% Here's the sentence referencing the Proctor paper from 2020bc.tex: 
% Proctor~\cite[Theorem~5BC]{ProctorBruhat} showed that the Bruhat order on $\bn$ is isomorphic to a similar order on tableaux $\{B(w) \,|\, w \in \bn \}$ of shape $(n,n-1,\dotsc,1)$ defined by placing the increasing rearrangement of $w_i \cdots w_n$ in row $i$, for $i = 1, \dotsc, n$.
%($\star$ Introduce and polish the following lemma about length and inversions.  Actually, if we can find this in a published paper,
%then we can state it without proof and cite the published paper.  It's definitely known.  Try looking on Google or MathSciNet.) This is in 
% \begin{proof}
%   First note that $\forall w \in \bn, w_1 \cdots w_n$ is the short one-line notation for an element of $\mfb{n-1}$ with the letter $n$ or $\ol n$ inserted somewhere. ($\star$ Rewritten: First note that the short one-line notation of $w$ contains either $n$ or $\ol n$.) When $n=1$, $\bn$ contains two elements with short one-line notations $1$ and $\ol 1$. 
%   In the first case, \begin{equation*}
%       \ell(w)=0=0+0=\inv(1)+\sumsb{i\\w_i < 0} |w_i|.
%   \end{equation*}
%   In the second case, \begin{equation*}
%       \ell(w)=1=0+1=\inv(\ol 1)+\sumsb{i\\w_i < 0} |w_i|.
%   \end{equation*}
%   Suppose that \begin{equation*}
%     \forall w \in \bn \; \ell(w) = \inv(w_1 \cdots w_n) + \sumsb{i\\w_i < 0} |w_i|.
%   \end{equation*}
%   Then in $\mfb{n+1}$, there are 2 cases:\\
%   \\
%   Case 1: $n+1$ is in the short one-line notation at position $i$. Then since $n+1$ is the largest letter in the short one-line notation, it increases $\inv(w)$ by $n+1-i$, $\sumsb{i\\w_i < 0} |w_i|$ by $0$, and $\ell(w)$ by $n+1-i$, so the claim holds for this case.\\
%   Case 2: $\ol{n+1}$ is in the short one-line notation at position $i$. Then since $\ol{n+1}$ is the smallest letter in the short one-line notation, it increases $\inv(w)$ by $i-1$, $\sumsb{i\\w_i < 0} |w_i|$ by $n+1$, and $\ell(w)$ by $n+i$, so the claim holds for this case as well.\\
%   \\
%   Thus, the claim holds. ($\star$ Proof may need some rewriting. Have we fully justified the increase to $\ell(w)$ in both cases? Should we hold constant some $w \in \bn$? What is "it" when we say "it increases $\inv(w)$ by..."?)
% \end{proof}
%For $W = \sn$ or $\bn$ and $J$ a %subset of generators, define 
The first equality in (\ref{eq:wejmdefns}) implies that when 
$J = J_{[a,b]}$,
%$W = \bn$, 
the subset $\Wejm$ of $\bn$ may be characterized in terms
of long 
%or short 
one-line notation as follows.  
%Specifically, we have the following.

% ($\star$ Maybe make a lemma out of this.)

% \begin{equation*}
%     \Wejm = \{ w \in W \,|\, sw^{-1} > w^{-1} \text{ for all } s \in J \}.
% \end{equation*}
% This means that $\Wejm$ is the set of elements $w \in \bn$ such that for all $s_i \in J$, the long one-line notation $w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n$ of $w$ satisfies
% \begin{enumerate}
%     \item $\ol 1$ appears before $1$ if $i = 0$
%     \item $i$ appears before $i+1$ (and $\ol{i+1}$ before $\ol i$) if $i \geq 1$.
% \end{enumerate}
% This also means that $\Wejm$ is the set of
% elements $w \in \bn$ such that for all $s_i \in J$, the short one-line notation $w_1 \cdots w_n$ satisfies
% %\begin{lem}
% %    Fix subset $J$ of generators of $\bn$.  We have that $\Wejm$ is the subset of elements $w \in \bn$
%     %$w = w_1 \cdots w_n$ in $\bn$ with the property 
% %    such that for each generator $s_i \in J$ ($i \geq 0$),
% %\begin{equation*}
% %    \begin{cases}
% \begin{enumerate}
% \item If $i = 0$ then $1$ appears in $w_1 \cdots w_n$,
% \item If $i > 0$ then one of the following is true:
% \begin{enumerate}
%     \item $i$, $i+1$ appear in $w_1 \cdots w_n$ with $i$ earlier,
%     \item $\ol i$, $i+1$ appear in $w_1 \cdots w_n$,
%     \item $\ol i$, $\ol{i+1}$ appear in $w_1 \cdots w_n$ with $\ol{i+1}$ earlier.
%     \end{enumerate}
% \end{enumerate}

% %($\star$ Rewrite the following, replacing two-sided cosets with one-sided cosets of the form $wW_J$ and make it clear that it holds for $W = \sn$ or $\bn$.
% %Introduce the notation $W/W_J$ for the set of such cosets. $\star$ Need to pick consistent language/notation for posets: $<$ vs. $\leq$ ? To be consistent with 2020bc.tex (Section 2.3), when we talk about the Bruhat order, let's use $\leq$. Also, mention $\leq_W$ if there is any danger of ambiguity.)
% %For any subset $J$ of generators of $W$, the 

% %Minimum coset representatives
% In the special case that $J = \{ s_a, \dotsc, s_b\}$ with $0 \leq a < b$, formulas come out like this:

% If $a \geq 1$ then $\Wejm$ is the subset of elements $w$ in $\bn$ such that letters $a, \dotsc, b+1$ appear in increasing order in the long one-line notation of $w$, equivalently,
% \begin{equation*}
% w_a^{-1} < \cdots < w_{b+1}^{-1}.
% \end{equation*}
%   If $a = 0$ then $\Wejm$ is the subset of elements $w$ of $\bn$ such that letters $-1, 1, \dotsc, b+1$ appear in increasing order in the long one-line notation of $w$,
%   equivalently,
% \begin{equation*}
% w_{\ol 1}^{-1} < w_1^{-1} < \cdots < w_{b+1}^{-1}.
% \end{equation*}

%$\star$ I think that condition (2) in the following lemma is not valid unless $a=0$.
%($\star$ I agree. Added the extra condition.)

%\begin{lem}
%   , the long one-line notation
%    of $w^{-1}$ contains subwords
    %contains the subword
    %\begin{equation*}
    %\begin{cases} 
    %    \ol b \cdots \ol a \cdot a \cdots b &\text{if $a > 0$},\\
    %    \ol b \cdots \ol 1 1 \cdots b
    %    &\text{ if $a = 0$}.
    %    \end{cases}
    %\end{equation*}
%    Equivalently, the long one-line notation of $w^{-1}$ satisfies
%    \begin{enumerate}
%        \item $w_{\ol b}^{-1} < \cdots < w_{\ol a}^{-1} < w_a^{-1} < \cdots < w_b^{-1}$, if $a > 0$
%    \end{enumerate}
%    \end{equation*}
    %\end{lem}

\begin{lem}\label{l:minrepprop}
Given reversal $s_{[a,b]} \in \bn$ and
corresponding generators $J = J_{[a,b]}$, 
%indices $a$, $b$ with $0 \leq a < b < n$ and
%a 
%subinterval $[a,b]$ of $[0,n-1]$ and 
%the 
%corresponding 
%generator 
%subset 
%generator subset $J = \{ s_a, s_{a+1}, \dotsc, s_b \}$ 
%of $\bn$,
%of generators 
%of $\bn$,
    each minimum-length coset representative $w \in \Wejm$ satisfies
%In $\mfb{n}$, the minimum length coset representatives $w \in \Wejm$ have the following property:  If $J = \{s_a, \cdots, s_b\}$ for $b>a \geq 0$, then we have

  $(1)$ %$w^{-1}_{\ol{i+1}} < w_{\ol i}^{-1}$ and 
    $w_i^{-1}<w_{i+1}^{-1}$ 
for $i = \max \{1,a\}, \dotsc, b-1$, and
%TODO - claim that \ref{l:minrepprop}, $\mathrm{min}\{\ol{1}, \ol{a_k}\}$ to $\ol{b_k}$ are in order (Proven in Case (1b)

$(2)$ $w_{\ol{1}}^{-1} < w_1^{-1}$ if $a=\ol b$.

% ($\star$ Could alternatively write:
% $\forall \ i \in [\mathrm{max}\{1,a\}, b-1]$.)
% $w \in \Weim $ satisfies that the paths ending in positions $a$ to $b+1$ are. %something$
\end{lem}
%($\star$ Observation: when $i<0$, also have $w_{i+1}^{-1}<w_{i}^{-1}$.)

\begin{proof}
Omitted.
% By 
% (\ref{eq:bruhatinverse}) and
% the action preceding (\ref{eq:s2nrearrange}), we have
% %( Observe that for $i \in [0,n-1]$ we have 
% the equivalences
% %The two integer inequalities in (1) are equivalent to the Bruhat inequality 
% \begin{equation*}
%     ws_i > w
%     \ \Longleftrightarrow\
%     s_i w^{-1} > w^{-1}
%     \ \Longleftrightarrow\
% %$s_iw^{-1} > w^{-1}$, 
% %i.e., $ws_i > w$.
% \begin{cases}
%     %w_{\ol{i+1}}^{-1} < w_{\ol i}^{-1} \text{ and } 
%     w_i^{-1}<w_{i+1}^{-1}
%     &\text{if $1 \leq i \leq n-1$},\\
%     w_{\ol1}^{-1} < w_1^{-1}
%     &\text{if $i = 0$}.
% \end{cases}
% \end{equation*}
% Thus by (\ref{eq:jabdef}) and (\ref{eq:wejmdefns}),
% %we have that 
% the combination of $(1)$, $(2)$
% is equivalent to the minimality of $w$ in 
% $wW_J$.
% %$\Wejm$.
\end{proof}


% Proof by contradiction.  
% % i > 0 case
% Case (1a): For $w \in \mfb{n}$, suppose $w_i^{-1} > w_{i+1}^{-1}$ 
% for some $i \in [1, b-1]$.
% Left multiplying $w$ by $s_i$ swaps $w_i$ and  $w_{i+1}$.
% Thus we have $(s_i w^{-1})_i < (s_i w^{-1})_{i+1}$.
% Then we have $s_i w^{-1}<w^{-1}$. Equivalently, we have $w s_i<w$, contradicting to the fact that $w\in \Wejm$. 

% Case (1b): For $w \in \mfb{n}$, suppose $w_i^{-1} > w_{i+1}^{-1}$ 
% for some $i \in [\overline{b},\overline{2}]$.
% Then note that if $w_i^{-1} < w_{i+1}^{-1}$,
% then $w_{\overline{i}}^{-1} > w_{\overline{i+1}}^{-1}$, which contradicts Case (1a). 

% % Consider case for i < 0 - use symmetry

% % i = 0 case
% Case (2): Suppose $w_{\ol{1}}^{-1}>w_1^{-1}$. Then $s_0w^{-1}<w^{-1}$, i.e., $ws_0<w$, contradicting to the fact $w\in \Wejm$. 
% % Does Case 2 need a sentencing demonstarting that $w_{\ol{1}}^{-1}>w_1^{-1}$ implies $s_0 \in I$?
% \end{proof}




  
%  %  %  PREVIOUS VERSION OF PARAGRAPH TEXT:
% For any subsets $I$, $J$ of generators of $W$,
%   the Bruhat order on $W$ induces a poset structure on
%   %$W_{\iota(M)} \backslash W / W_{\iota(O)}$
%   $W_I \backslash W / W_J$
%   as follows.  We declare
% $W_IvW_J \leq W_IwW_J$ if elements of the cosets satisfy any of the
% three (equivalent) inequalities in the Bruhat order on $\mfs r$~\cite[Lemma~2.2]{DougInv}.
%   \begin{enumerate}[(i)]
%       \item The minimal element of $W_IvW_J$ is less than or equal to
% the minimal element of $W_IwW_J$.
%       \item The maximal element of $W_IvW_J$ is less than or equal to
% the maximal element of $W_IwW_J$.
%       \item At least one element of $W_IvW_J$ is less than or equal to
% at least one element of $W_IwW_J$.
%   \end{enumerate}
% %  (See \cite{DougInv}.)
%   We call this poset the {\em Bruhat order on
%   %the double cosets
%   $W_I \backslash W / W_J$}.


%($\star$ After taking appropriate material out of the following section and rewording it to complete the above tasks, delete or comment out the following section.  It's already copied as Section 8 at the end of the document, just in case we need anything.)

% Let $W^J_-$ be the set of all elements $w \in W$ satisfying 
% %sw < w$ for all $s \in I$ and 
% $ws > w$ for all $s \in J$. 

%We present some properties of minimum length coset representatives in $W$ that will be useful in the proof of our main result.


\section{The Hecke algebra of the hyperoctahedral group}\label{sec: Hecke algebras}

%($\star$ Can we get away with defining only the Hecke algebra of the hyperoctahedral group?  Should we combine this with the previous section?)

%($\star$ If we use just the hyperoctahedral group Hecke algebra,
%should we 


%Work on main results first and then come back to this point.)

Given Coxeter group $W$ with generator set $S$, define
the {\em Hecke algebra} $H = H(W)$ of $W$ to be
the $\zqq$-span of $\{ T_w \,|\, w \in W \}$
with 
%multiplicative 
unit $T_e$ and multiplication defined by
\begin{equation}\label{eq:genlhecke}
%  \begin{cases}
  T_wT_s =
  \begin{cases}
    qT_{ws} + (q-1)T_w &\text{if $ws < w$},\\
    T_{ws} &\text{if $ws > w$},
  \end{cases}
%    T_uT_v = T_{uv} &\text{if $\ell(uv) = \ell(u) + \ell(v)$}.
%        T_uT_v = T_{uv} &\text{if $\ell(uv) = \ell(u) + \ell(v)$}.
%  \end{cases}
\end{equation}
where $s \in S$ and $w \in W$.
% , and $<_W$ is the Bruhat order on $W$. ($\star$ We already define the Bruhat order on $W$ in the first sentence of section 1 - is this redundant? I guess this is to emphasize that we are using the \emph{strictly less than} part of the Bruhat order, not weakly less than. )
This formula guarantees that for $w \in W$ and
any reduced expression $s_{i_1} \cdots s_{i_\ell}$ for $w$,
%a reduced expression,
we have $T_w = \smash{T_{s_{i_1}} \cdots T_{s_{i_\ell}}}$.
Call $\{ T_w \,|\, w \in W \}$ the {\em natural basis} of $H$.
%$H(W)$.
It is easy to see that the specialization of $H$
%$H(W)$ 
at $\smash{\qp12 = 1}$
is isomorphic to $\mathbb Z[W]$.
% ($\star$ Present the type-$\msfBC$ Hecke algebra in terms of generators and relations, just like 
% %in a way parallel to what 
% we did for the hyperoctahedral group in (\ref{eq:bnpresentation}) We can use this in place of or in addition to the above description.)
% ($\star$ From this point on, use $\hbnq$ instead of $\hnq$ whenever we work with the type-$\msfBC$ Hecke algebra, i.e., probably all of the time.)
A second basis~\cite{KLRepCH} 
of $H$ is the (rescaled)
%, introduced in 
{\em Kazhdan--Lusztig basis} $\{ \wtc wq \,|\, w \in W \}$,
related to the natural basis by
\begin{equation}\label{eq:wtcdef}
  \wtc wq = \sum_{v \leq w} P_{v,w}(q) T_v,
\end{equation}
where
%$\leq_W$ is the Bruhat order on $W$ and 
$\{ P_{v,w}(q) \,|\, v, w \in W \} \subseteq \mathbb Z[q]$ are the
%recursively defined 
%(type-$\msfA$)
{\em Kazhdan--Lusztig polynomials},
whose recursive definition appears in ~\cite{KLRepCH}.
%($\star$ Include Schubert interpretation of coefficients of $P_{v,w}(q)$.
%($\star$ For more details see
% \cite[Theorem~6.1.19, Section 13.2]{BilleyLak}?)
Coefficients of these polynomials may be interpreted in terms of
intersection cohomology $\icoh^*(\schub w)$~\cite{KLSchub}, where $\schub w$ is a certain {\em Schubert variety} indexed by $w$. (See, e.g., \cite{BilleyLak}.) In particular, %Specifically,
when $\schub w$ is rationally smooth,
%and $v \leq_W w$
%When $W = \sn$ and $w\in \sn$ \avoidsp,
%it is well known that
%all of
%each polynomial $P_{v,w}(q) = 1$ for all $v \leq_W w$~\cite[Theorem\,A.2]{KLRepCH}.
all
%of the
polynomials
$\{P_{v,w}(q) \,|\, v \leq w\}$
%each of these polynomials is identically
are identically
$1$ \cite[Theorem\,A.2]{KLRepCH}.



%($\star$ Lakshmibai's result, specifically mentioning \theps) we have the following.
%($\star$ Can this setup paragraph to \Cref{p:abcsmoothkl} be simplified?)
%($\star$ Or put this after the proposition?)
%\cite{LakSan}.
%When $W = \bn$ we have the following analogous result.
%holds when $W = \bn$.
% \begin{prop}
% %[{\cite[Proposition~3.3]{SkanHyperGC}}]
% \label{p:abcsmoothkl}
%   For $W$ equal to $\sn$ or $\bn$ and $w \in W$ \avoidingp,
%   the Kazhdan--Lusztig basis element $\wtc wq$
%   of $H(W)$
%   %$\hbnq$
%   satisfies
%   %we have
%   \begin{equation*}
%     \wtc wq = \sum_{v \leq_W w} T_v.
% %    \wtc wq = \sum_{v \leq_{\bn} w} T_v.
%   \end{equation*}
% \end{prop}
% \begin{proof}
% % By ($\star$ find Lakshmibai result), the stated pattern avoidance implies ($\star$ is equivalent to?) rational smoothness ($\star$ or just smoothness? Is this included in the proofs of Propositions 3.1 -- 3.2 in 2020bc.tex? This is just Proposition~3.3 in 2020bc.tex. Maybe just reference it.)
% % %$\star$ Just cite Lakshmibai and refer reader to Billey--Lakshmibai for 
% % %  Omitted. 
% % %  By \Cref{p:bcavoid},  the
% % %  Schubert variety
% % %  $\Bschub w$
% % %  is smooth and therefore rationally smooth.
% % %  Then by \cite[Theorem\,A.2]{KLRepCH} we have $P_{v,w}(q) = 1$
% %   %  for all $v \leq w \in \bn$.
% Omitted.
% \end{proof}
% $\star$ Introduce this.


%$\star$ State any facts about products in the Hecke algebra that we will need and which don't involve path families.

% \begin{prop}\label{prop: algebraic product formula}
%   ($\star$ This can probably be omitted. This is referenced in Theorems~\ref{t:Awiringdiagramprod} and~\ref{t:BCwiringdiagramprod}, which we won't need anymore.)
%   Suppose that
%   \begin{equation}\label{eq:Ashorttpluseprod}
%     \ewtc{s_{i_1}}q \cdots \ewtc{s_{i_{k-1}}}q = \sum_{w \in \sn} a_w T_w
%   \end{equation}
%   for some polynomials $a_w = a_w(q)$ in $\mathbb Z[q]$,
%   and let $s$ 
%   %= s_{i_k}$ 
%   be any
%   adjacent transposition.  Then we have
%   \begin{equation}\label{eq:Atpluseprod}
% %    \begin{aligned}
%     \ewtc{s_{i_1}}q \cdots \ewtc{s_{i_{k-1}}}q \ewtc sq
% %    = \sumsb{w \in \sn\\ws>w} (a_w + qa_{ws}) T_w
% %    + \sumsb{w \in \sn\\ws<w} (a_{ws} + qa_w) T_w\\
%     = \sumsb{w \in \sn\\ws>w} (a_w + qa_{ws}) (T_w + T_{ws}).
% %    \end{aligned}
%   \end{equation}
% \end{prop}
% \begin{proof}
% %By definition, note that if $ws>w$, $T_wT_s= T_{ws}$, and if $ws<w$, $T_w T_s = (q-1)T_w + q T_{ws}$. Thus 
% By (\ref{eq:Ashorttpluseprod}), the left-hand side of (\ref{eq:Atpluseprod}) is
% \begin{equation*}
%     \begin{aligned}
% %        \ewtc{s_{i_1}}q \cdots \ewtc{s_{i_{k-1}}}q \ewtc sq  &= \left( \sum_{w\in\sn} a_w T_w \right) (T_e+T_s)
% %        \\ &= 
% \Big( \ntksp \sumsb{w\in\sn \\ws>w} a_w T_w + \sumsb{w\in\sn \\ws<w} a_w T_w \Big) (T_e + T_s)
% %&=  \sumsb{w\in\sn \\ws>w} \left( a_w T_w + a_w T_w T_s \right) + \sumsb{w\in\sn \\ws<w} \left( a_w T_w +a_w T_w T_s \right) 
%     &= \sumsb{w\in\sn \\ws>w} \left(a_w T_w + a_w T_{ws} \right) + \sumsb{w\in\sn \\ws<w}\left(qa_wT_w + qa_w T_{ws} \right)
%         \\ &= \sumsb{w\in\sn \\ws>w} (a_w+qa_{ws})T_w + \sumsb{w\in\sn \\ws<w} (qa_w + a_{ws})T_w,
%     \end{aligned}
% \end{equation*}
% which equals the right-hand side of (\ref{eq:Atpluseprod}).
% %which is exactly the first line of the proposition. By taking the first sum as is, and replacing $w$ with $ws$ and vice versa on the second sum, we trivially get
% %$$ = \sumsb{w \in \sn\\ws>w} (a_w + qa_{ws}) (T_w + T_{ws})$$
% %as desired.
% \end{proof}


%\begin{proof}
  %Omitted.  
%  $\star$ Do this.
%\end{proof}

% \ssec{Put this somewhere}
% %If $\sigma$ is a coset of the form $uW_J$, then define
% Define 
% \begin{equation*}
% T_{uW_J} \defeq \sum_{v \in uW_J} T_v.
% \end{equation*}

For each subset $J$ of generators of $W$, one forms a natural $\zqq$-submodule 
of $H$
%$\hbjq$ 
%of $\hbnq$ 
by taking the span of sums
%$W = \sn$ or $\bn$ and subset $J$ of the generators, 
%let $W_J$ be the subgroup generated by $J$
% For each interval $[a,b]$, let $J_{[a,b]} = \{s_a, \dotsc, s_{b-1}\}$.
%and 
%define
\begin{equation}\label{eq:cosetsum}
T_{uW_J} \defeq \ntksp \sum_{v \in uW_J} \ntksp T_v
\end{equation}
of natural basis elements of $H$
%$\hbnq$ 
with each sum corresponding to
%indices belonging to each 
a coset of $W_J$. This submodule
\begin{equation}\label{eq:hbjqdef}
H_J
%\hbjq 
\defeq \spn_{\zqq} \{ T_{uW_J} \,|\, u \in \Wejm \}
\end{equation}
in fact forms a left $H$-module.
%$\hbnq$-module. %($\star$ cite?).
%is closed under the action of left multiplication 
%For fixed $J$, these {\em coset sum} elements of Hecke algebra span a left-$\hbnq$-module, i.e., a subspace
%which is closed under the action of left multiplication by $\hbnq$.  
%($\star$ Mention that as $w$ varies over minimal coset representatives the resulting coset sums form a basis.) Call this module $H_J(q)$.
A nice formula for the action of $H$
%$\hbnq$ on $\hbjq$ 
on $H_J$ was given by Douglass~\cite[Proposition~2.3]{DougInv}. %the left multiplication rule for \eqref{equ: parabolic}.
\begin{prop}\label{p:douglass}
    Let $s$ be a generator of $W$, and let $J$ 
    %$W = \sn$ or $\bn$, let $J$ 
    be some subset of the generators. %and let $s$ be a generator of $W$. 
    Then for all $w \in  W$, we have
\begin{equation}\label{equ:product rule} T_s T_{w W_J} = \left\{ \begin{array}{cc}
    q T_{s w W_J} + (q-1)T_{w W_J} & \text{if } swW_J < wW_J, \\
    T_{s w W_J} & \text{if } swW_J > wW_J ,\\
    q T_{w W_J} & \text{if } swW_J = wW_J .\end{array} \right.
    \end{equation}
\end{prop}

% Further, we have the following two corollaries and a proposition:
% ($\star$ Say something more descriptive.)
%It follows that we have a very simple formula for the product of $T_vT_{W_J}$ in the case that $v$ belongs to the
%y an element $T_v$ with $v$ belonging to the subgroup $W_J$. 
%($\star$ Is this corollary necessary?  Or can we just refer to the above proposition when needed?) 
%, we have a very simple multiplication rule.
%is quite simple. 

%($\star$ Redo this discussion and the results following it so that they are all consistent with (\ref{eq:jabdef}).
\begin{cor}\label{c:doug}
    If $v \in W_J$, then
%\begin{equation}\label{eq:dougcor}
    $T_{v}T_{W_J} = q^{\ell(v)} T_{W_J}$.
%\end{equation}
   \end{cor}
% \begin{proof}
%     Since $v \in W_J$, by definition, $v=s_{i_1} s_{i_2} \cdots s_{i_{\ell(v)}}$, where each $s_{i_k} \in J$. Further, we know for any $s\in J_{[a,b]}$, $s W_J=W_J$. Therefore, 

%     $$ T_v T_{W_J} = T_{s_{i_1}} T_{s_{i_2}} \cdots T_{s_{i_{\ell(v)}}} T_{W_J} = q^{\ell(v)} T_{W_J} $$
%     by $\ell(v)$ uses of \Cref{p:douglass}.
% \end{proof}


%\section{Products corresponding to 
%%more general 
%star networks}\label{sec:products of star networks}

%$\star$ We can probably get away with $W = \bn$ only.  Place as many of the following results in the Hecke algebra section, as long as we can state and prove them without using planar networks. $\star$ See comment.
% TJP - I think the entire remaining section, which is now just prop 5.1 and cor 5.2, can be moved to section 3.


%$\star$ Need to define $\wtc{s_{[a_i,b_i]}}q$ here. -NOTE- we define $\wtc{w}q$ in \Cref{define_wtc}, should this be moved to section 5? 

%$\star$ Since this proposition is already in Section 5, do we want to move this and the following proposition to Section 3, right after we introduced \emph{reversals}? Do we need star networks here? 

% ($\star$ Introduce, state and prove a type-$\msfBC$ analog of this.  Is it basically \Cref{p:starprod}? Yes.)

% \begin{prop}
%   Suppose that
%   \begin{equation*}
%     \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k-1},b_{k-1}]}}q = \sum_{w \in \sn} c_w T_w
%   \end{equation*}
%   for some polynomials $c_w = c_w(q)$ in $\mathbb Z[q]$, and let $[a,b]$ be any subinterval of $[1,n]$.  
%   Let $W = \sn$ and let $I = \{ s_{a}, \dotsc, s_{b-1}\}$.
% %  and let $s = s_{i_k}$ be any
% %  adjacent transposition.  
% Then we have
%   \begin{equation*}
%     \begin{aligned}
%     \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k-1},b_{k-1}]}}q \wtc{s_{[a,b]}}q  &=
%     \sum_{w \in W^I_-}
%     %\\wv>w\\\text{for all}\\{v \in \mfs{[a_k,b_k]}}} 
%     \Big(\nTksp\sum_{v \in \mfs{[a,b]}} \ntksp q^{\ell(v)}c_{wv} \Big) T_w \wtc{s_{[a,b]}}q.
%     \end{aligned}
%   \end{equation*}
% \end{prop}

% ($\star$ Do we need to define S in the following proposition?)

% ($\star$ We should. Probably not here, but in Section~\ref{s:bn} when we first defined $\bn$.)
Let us consider the module $\hbjq \defeq H_J$ in the special case that $H = \hbnq$
and $J = J_{[a,b]}$, and its relation to Kazhdan--Lusztig basis elements.  
%Call this module $\hbjq$. 
By \cite[Section 13.3.7]{BilleyLak},
rational smoothness of Schubert varieties 
%is implied in 
of types $\msfA$, $\msfBB$, $\msfC$ 
is implied by pattern avoidance.
%by avoidance of \theps.
In particular, when $w \in \bn$ \avoidsp,
%these patterns, 
we have the stronger property
that type-$\msfBB$ and $\msfC$ Schubert varieties indexed by $w$ are both smooth.
%avoidance of the
Since a reversal 
%$w$ 
in 
%$\sn$ or 
$\bn$ avoids both of these patterns,
%$3412$ and $4231$, 
we have the following.
% a special case of \Cref{p:abcsmoothkl}.
\begin{prop}\label{p:wtcreversal}
    For each reversal $s_{[a,b]} \in \bn$,
    %$W$ equal to $\sn$ or $\bn$ and $w \in W$ a reversal,
    we have
\begin{equation*}
\wtc{s_{[a,b]}}q = \sum_{v\leq s_{[a,b]}} T_v.     
    \end{equation*}
\end{prop}
%($\star$ Introduce this. Something like the following.  
%Observe that \
Thus
%when $w \in \bn$ is the reversal $s_{[a,b]}$, the 
each Kazhdan--Lusztig basis element %$\wtc wq$ (\ref{eq:wtcdef}) 
indexed by a reversal $s_{[a,b]}$ is itself a coset sum (\ref{eq:cosetsum}) for the parabolic subgroup generated by $J = J_{[a.b]}$:
%$\{ s_a, \dotsc, s_{b-1}\}$,
\begin{equation}
T_{W_J} = 
    \wtc{s_{[a,b]}}q.
    %= T_{W_J}.
\end{equation}
By (\ref{eq:Jfactor}) and (\ref{equ:product rule}), other defining basis elements (\ref{eq:hbjqdef}) of $\hbjq$ can be written as
\begin{equation}\label{eq:TuWJ}
    T_{uW_J} = T_u \wtc{s_{[a,b]}} q,
\end{equation}
for $u \in \Wejm$.
It follows that each product of the form
    \begin{equation}\label{eq:klprod}
        \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k},b_{k}]}}q 
        %= \sum_{u \in W} c_u T_u,
    \end{equation}
belongs to 
%$\hbjq$ 
%or 
$H^{\msfBC}_{J_{[a_k,b_k]}}$ 
%for $J = \{ s_{a_k}, \dotsc, s_{b_k-1}\}$, 
and we can expand it
in the defining basis of this module %$\hbjq$ 
as follows.
%By \Cref{p:douglass}
\begin{prop}\label{p:klprodcosetexpansion} 
Suppose that the product of the first $k-1$ factors of (\ref{eq:klprod})
expands in the natural basis of $\hbnq$
as
%For $W = \sn$ or $\bn$ with generator set $S$, define a subset of generators $J:= \{s_{a_k}, \dotsc, s_{b_k-1}\}$. Let $W_J$ be the subgroup generated by $J$. Suppose that
\begin{equation}\label{eq:klprodkminus1}
        \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k-1},b_{k-1}]}}q = \sum_{v \in \bn} c_v T_v,
    \end{equation}
    for some polynomials $\{ c_v = c_v(q)\,|\, v \in \bn \}$ in $\mathbb Z[q]$,
    and define $J = J_{[a_k,b_k]}$.
    %$J = \{ s_{a_k},\dotsc, s_{b_k-1}\}$.
    %, and let $[a,b]$ be any subinterval of $[1,n]$. % If we use only W=Bn, we could define 'type-BC' intervals here.
Then the full product (\ref{eq:klprod})
expands in the defining basis of $\hbjq$ as
\begin{equation}\label{eq:klprodcosetexpansion}
%    \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k-1},b_{k-1}]}}q\cdot \wtc {s_{[a_k,b_k]}} q = 
\sum_{w \in \Wejm} \ntksp \bigg( \sum_{u \in  W_{J}} \ntnsp q^{\ell(u)} c_{wu} \bigg) T_w \wtc {s_{[a_k,b_k]}} q.
\end{equation}
\end{prop}
\begin{proof}
Using
(\ref{eq:Jfactor}) to factor each element $v$ on the right-hand side of 
(\ref{eq:klprodkminus1}) as $v = wu$ with $w \in \Wejm$, $u \in W_J$,
%(\ref{eq:Jfactor})
we may express the product (\ref{eq:klprod}) as
%Observe that 
\begin{equation*}  
%\begin{aligned}
%\wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_{k-1},b_{k-1}]}}q\cdot \wtc {s_{[a_k,b_k]}} q &= 
%\bigg(\sum_{u \in \bn} \ntksp c_u T_u \bigg)
%\cdot 
%\wtc{s_{[a_k,b_k]}}q =
%&
\sum_{w \in \Wejm} \ntksp \bigg( \sum_{u \in  W_{J}} \ntksp c_{wu} T_w T_u \bigg) \wtc 
 {s_{[a_k,b_k]}} q.
 %\end{aligned}
\end{equation*}
%as every element $u \in W$ can be expressed as $wv$, with $w \in \Wejm$ and $v \in W_J$. 
Then by \Cref{c:doug}, 
we have
%each product 
$T_u \wtc{s_{[a_k,b_k]}}q
%$ equals $
=
q^{\ell(u)}\wtc{s_{[a_k,b_k]}}q$ and 
%we have
the claimed formula.
%\begin{equation*}  
%    T_{wv} \wtc {s_{[a_k,b_k]}} q = q^{\ell(v)} T_w \wtc {s_{[a_k,b_k]}} q
%\end{equation*}
%($\star$ maybe just write $T_{wv}=q^{\ell(v)}T_w$)
%and thus
% \begin{equation*}  
% \sum_{w \in \Wejm} \left( \sum_{v \in  W_{J}} c_{wv} T_{wv} \right) \wtc {s_{a_k,b_k}} q =  \sum_{w \in \Wejm} \left( \sum_{v \in  W_{J}} q^{\ell(v)} c_{wv} \right) T_w \wtc {s_{a_k,b_k}} q
% \end{equation*}
% as desired.
\end{proof}


\section{Type-\texorpdfstring{$\mathsf{BC}$}{BC} star networks}\label{s:BCstarnets}

%($\star$ Introduce this by saying something about planar networks in type-$\msfA$.)
To graphically represent products of %Kazhdan--Lusztig basis elements
the form (\ref{eq:klprod}),
we extend the idea of type-$\msfBC$ wiring
diagrams to include planar networks in which
%We can generalize the idea of wiring diagrams in types $\msfA$ and $\msfBC$ by allowing 
interior vertices may have indegrees and outdegrees greater than 2.
% ($\star$ Keep in mind that there is no definition of type-$\msfBC$ planar network in general; we need the type-$\msfBC$ stars to make things work.)
% ($\star$ Define type-$\msfBC$ simple star networks as in 2020bc.tex, except without using the type-$\msfA$ definition to do it.)
In particular, we associate
%We begin this extension by associating 
to each reversal $s_{[a,b]} \in \bn$ 
%defining
%In particular, we define 
a {\em type-$\msfBC$ simple star network} $\smash{F_{[a,b]}}$ having $2n$ source vertices on the left and $2n$ sink vertices on the right, both labeled $\ol n, \dotsc, n$ from bottom to top.
%to be a planar network corresponding to the reversal $s_{[a,b]} \in \bn$, denoted $F^{[\ol n, n]}_{[a,b]} = $F_{[a,b]} = F_{s_{[a,b]}}$.
%Recall that
% for a  type-$\msfBC$ satisfactory interval $[a,b]$,
%we have 
For the three cases $a=b$, $a = \overline{b}$ $(b > 0)$, $0 < a < b$ of reversals,
we include edges and $0$, $1$, or $2$ additional
interior vertices as follows.

% ($\star$ Do we need to include the trivial case: $a=b$ and $[a,b]=\emptyset$?)
% Could be useful to define a 'type-BC interval' so that these two cases do not need to be repeatedly references
\begin{enumerate}
\item When $\smash{F_{[a,a]} = F_\emptyset}$ has a directed edge from source $i$ to sink $i$, for $i = \overline{n},\dotsc,n$. 
\item When $a = \overline{b}$ $(b > 0)$, $F_{[a,b]}$ has one interior vertex. For $i = a,\dotsc, b$, a directed edge begins at source $i$ and terminates at the interior vertex, and another directed edge begins at the interior vertex and terminates at sink $i$.
\item When $0 < a < b$, $F_{[a,b]}$ has two interior vertices. For $i = a, \dotsc, b$, a directed edge begins at source $i$ and terminates at the upper interior vertex, and another directed edge begins at the upper interior vertex and terminates at sink $i$. For $i = \overline{b}, \dotsc, \overline{a}$, a directed edge begins at source $i$ and terminates at the lower interior vertex, and another directed edge begins at the lower interior vertex and terminates at sink $i$. 
\end{enumerate}
 
% ($\star$ Is there a difference between $F_{[a,b]}$ and $F'_{[a,b]}$? - No difference, though we should choose a consistent notation. $F'_{[a,b]}$ is used frequently in 2020bc.tex, while $F_{[a,b]}$ is used to represent a simple network in our main result. Also in our main result, $F'$ is the network composed of $k-1$ stars.)
%\noindent
For example, the seven type-$\msfBC$ simple star networks
%$F_{\emptyset} = F_{[1,1]} = F_{[2,2]} = F_{[3,3]}$ and $ F_{\smash{[\ol 1, 1]}}, F_{\smash{[\ol 2, 2]}}, F_{\smash{[\ol 3, 3]}}, F_{[1,2]}, F_{[2,3]}, F_{[1,3]}$,
\begin{equation}\label{eq:csimplestars}        
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,2.5);
\draw[-, very thick] (-.5,1.5) -- (.5,1.5);
\draw[-, very thick] (-.5,0.5) -- (.5,0.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-2.5);
\node at (0, -3.8) {$F_\emptyset$};
\end{tikzpicture}
\ntnsp,\quad
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,2.5);
\draw[-, very thick] (-.5,1.5) -- (.5,1.5);
\draw[-, very thick] (-.5,0.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,0.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-2.5);
\node at (0, -3.8) {$F_{[\ol1,1]}$};
\end{tikzpicture}
\ntnsp,\quad
%\ ,\;
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,2.5);
\draw[-, very thick] (-.5,1.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,0.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,0.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,1.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-2.5);
\node at (0, -3.8) {$F_{[\ol2,2]}$};
\end{tikzpicture}
\ntnsp,\quad
%\ ,\;
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,-2.5);
\draw[-, very thick] (-.5,1.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,0.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,0.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,1.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,2.5);
\node at (0, -3.8) {$F_{[\ol3,3]}$};
\end{tikzpicture}
\ntnsp,\quad
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,2.5);
\draw[-, very thick] (-.5,1.5) -- (.5,0.5);
\draw[-, very thick] (-.5,0.5) -- (.5,1.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-2.5);
\node at (0, -3.8) {$F_{[1,2]}$};
\end{tikzpicture}
\ntnsp,\quad
%\ ,\;
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,1.5);
\draw[-, very thick] (-.5,1.5) -- (.5,2.5);
\draw[-, very thick] (-.5,0.5) -- (.5,0.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,-0.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,-2.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-1.5);
\node at (0, -3.8) {$F_{[2,3]}$};
\end{tikzpicture}
\ntnsp,\quad
%\ ,\;
\begin{tikzpicture}[scale=.5,baseline=0]
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (0.9,2.5) {$\scriptstyle 3$};
  \node at (0.9,1.5) {$\scriptstyle 2$};
  \node at (0.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (0.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (0.9,-0.5) {$\scriptstyle {\ol 1}$};
\draw[-, very thick] (-.5,2.5) -- (.5,0.5);
\draw[-, very thick] (-.5,1.5) -- (.5,1.5);
\draw[-, very thick] (-.5,0.5) -- (.5,2.5);
\draw[-, very thick] (-.5,-0.5) -- (.5,-2.5);
\draw[-, very thick] (-.5,-1.5) -- (.5,-1.5);
\draw[-, very thick] (-.5,-2.5) -- (.5,-0.5);
\node at (0, -3.8) {$F_{[1,3]}$};
\end{tikzpicture}
\ntnsp,\quad
%\ ,\;
\ntksp
\end{equation}
correspond to the reversals
$s_\emptyset = e$, $s_{[\ol1,1]}$, $s_{[\ol2,2]}$, $s_{[\ol3,3]}$, $s_{[1,2]}$, $s_{[2,3]}$, $s_{[1,3]}$ in $\mfb3$.
%$\star$ Use $\net{BC}n$ or $\net{BC}n$ to denote the set of  concatenations of these.  
% (Will we need both ordinary $\circ$ and condensed $\bullet$ concatenations?)
Define $\net{BC}n$ to be the set of all 
%type-$\msfBC$ star networks, that is, 
concatenations
%\begin{equation}\label{eq:BCconcat}
$F = F_{[a_1,b_1]} \circ \cdots \circ F_{[a_k,b_k]}$
%\end{equation}
of type-$\msfBC$ simple star networks
and call these {\em type-$\msfBC$ star networks}.
%of the form $F^{[\ol n, n]}_{[a,b]}$.
% Do we need to explain a concatenation?
For instance, 
%in $\net{BC}3$ 
%we have 
the type-$\msfBC$ star network
%\begin{equation}\label{eq:starnetF}
$F = F_{[1,3]} \circ F_{[2,3]} \circ F_{[1,2]} \circ F_{[\ol1,1]} \in \net{BC}3$
is shown in (\ref{eq:pathfamex}).
%\end{equation}
%$F = F_{[1,2]} \circ F_{[2,3]} \circ F_{[\ol 1,1]} 
%\in \net{BC}3$ 

% \ssec{Type-$\msfA$ planar networks ($\star$ Can we remove this subsection?)}\label{ssec: type A}

% ($\star$ Check to see if some of these definitions need to be kept for the $\msfBC$ results.)
% %Define type-$\msfA$ star networks, including the special case of wiring diagrams.)

% % ($\star$ The diagram "eq:132313" referenced by this subsection is missing)

% For all $F \in \net A{[h,l]}$, the set $\Pi(F)$ associates an element of
% $\mathbb Z[\mfs{[h,l]}]$ to $F$:
% we say that $F$ {\em graphically represents}
% \begin{equation}\label{eq:Gtozsn}
% \sum_{\pi \in \Pi(F)} \type(\pi)
% \end{equation}
% {\em as an element of $\mathbb Z[\mfs{[h,l]}]$}.
% For example, the network $F$ in (\ref{eq:132313})
% \begin{equation}\label{eq:132313}
% F = F_{[1,3]} \circ F_{[2,3]} \circ F_{[1,3]} = 
% \begin{tikzpicture}[scale=.5,baseline=-10]
% \node at (-.4,0.5) {$\scriptstyle 3$};
% \node at (-.4,-0.5) {$\scriptstyle{2}$};
% \node at (-.4,-1.5) {$\scriptstyle{1}$};  
% \node at (3.4,0.5) {$\scriptstyle 3$};
% \node at (3.4,-0.5) {$\scriptstyle{2}$};
% \node at (3.4,-1.5) {$\scriptstyle{1}$};  
% \draw[-] (0,0.5) -- (1,-1.5) -- (2,-1.5) -- (3,0.5);
% \draw[-] (0,-0.5) -- (1,-0.5) -- (2,0.5) -- (3,-1.5);
% \draw[-] (0,-1.5) -- (1,0.5) -- (2,-0.5) -- (3,-0.5);
% \end{tikzpicture},\qquad
% \pi = 
% \begin{tikzpicture}[scale=.5,baseline=-10]
% \node at (-.4,0.5) {$\scriptstyle 3$};
% \node at (-.4,-0.5) {$\scriptstyle{2}$};
% \node at (-.4,-1.5) {$\scriptstyle{1}$};  
% \node at (3.4,0.5) {$\scriptstyle 3$};
% \node at (3.4,-0.5) {$\scriptstyle{2}$};
% \node at (3.4,-1.5) {$\scriptstyle{1}$};  
% \draw[-, very thick] (0,0.5) -- (1,-1.5) -- (2,-1.5) -- (3,0.5);
% \draw[-, thick, dashed] (0,-0.5) -- (1,-0.5) -- (2,0.5) -- (3,-1.5);
% \draw[-] (0,-1.5) -- (1,0.5) -- (2,-0.5) -- (3,-0.5);
% \end{tikzpicture}
% \end{equation}
% can be covered by
% $72$ different path families: $12$ of each type $w \in \mfs 3$.
% Thus it graphically represents
% %the element
% %\begin{equation*}
% %$\wtc {s_1s_2s_1}1 \wtc {s_2}1 \wtc {s_1s_2s_1}1  =
% $12\,\wtc{s_1s_2s_1}1$
% as an element of $\mathbb Z [\mfs 3]$.
% %\end{equation*}
% %($\star$ Cite some paper for product of KL basis elements = concatentation?)

% Again for all $F \in \net A{[h,l]}$,
% the set $\Pi(F)$ also associates an element of
% $H_{[h,l]}(q)$ to $F$.
% To describe this element explicitly, we first assume
% that $F$ is formed by some iteration of ordinary or condensed concatenatation
% of simple star networks $F_{[c_1,d_1]}, \dotsc, F_{[c_t,d_t]}$ with
% internal vertices $\vertex z_1, \dotsc, \vertex z_t$. Observe that
% the intersection of two source-to-sink paths $\PiBC_i$, $\PiBC_j$ in $F$
% must be a disjoint union of the above internal vertices of $F$
% and paths between these.
% We say that $\PiBC_i$ and $\PiBC_j$ {\em meet} at a vertex $\vertex x$
% if both paths contain $\vertex x$, and enter it via different edges.
% %The departure of
% %$\PiBC_i$ and $\PiBC_j$ from the component of $\PiBC_i \cap \PiBC_j$
% %containing $x$ may result in a crossing of the two paths.
% Given a path family $\pi$ covering $F$,
% define a {\em defect} of $\pi$ to be a triple $(\PiBC_i, \PiBC_j, k)$ with
% \begin{enumerate}
%   \item $i < j$,
%   \item $\PiBC_i$ and $\PiBC_j$ meet at vertex $\vertex z_k$ of
% %    (\ref{eq:bulletconcat})
%     $F$ after having crossed an odd number of times.
% \end{enumerate}
% Let $\dfct(\pi)$ denote the number of defects of $\pi$.
% (This definition from \cite{CSkanTNNChar}
% generalizes those of \cite{BWHex}, \cite{Deodhar90}.)
% %(We follow the definition of \cite{CSkanTNNChar}, See \cite{BWHex}, \cite{Deodhar90}.)
% %($\star$ Cite Deodhar90, Billey--Warrington.)
% We say that $F$ {\em graphically represents}
% \begin{equation}\label{eq:Gtozhnq}
% \sum_{\pi \in \Pi(F)} q^{\dfct(\pi)} T_{\type(\pi)}
% \end{equation}
% {\em as an element of $H_{[h,l]}(q)$}.
% For example,
% %consider
% %the star network
% %in $\net A{[1,3]}$
% %($\star$ change this, because concatenation $\circ$
% %is not included in star network definition?)
% the path family $\pi$ in (\ref{eq:132313})
% %$F_{[1,3]} \circ F_{[1,2]} \circ F_{[1,3]}$
% %in $\snet A{[1,3]}$ and
% %path family $\pi = (\PiBC_1,\PiBC_2,\PiBC_3,\PiBC_4)$,
% %\begin{equation*}
% %\end{equation*}
% satisfies $\dfct(\pi) = 3$: the defects are
% $(\PiBC_1, \PiBC_2, 2)$, $(\PiBC_1, \PiBC_3, 3)$, $(\PiBC_2, \PiBC_3, 3)$.
% It is possible to show that the network $F$ in (\ref{eq:132313})
% %can be covered by
% %$72$ different path families, and
% graphically represents
% %the element
% %\begin{equation*}
% %$\wtc {s_1s_2s_1}q \wtc {s_2}q \wtc {s_1s_2s_1}q  =
% $(1+q)^2(1+q+q^2) \wtc{s_1s_2s_1}q$
% as an element of $H_3(q)$.

% It is clear that if $F$ graphically represents
% $D(q) \in H_{[h,l]}(q)$ as an element of $H_{[h,l]}(q)$, then
% it graphically represents
% $D(1)$ as an element of $\mathbb Z[\mfs{[h,l]}]$.
% Given a planar network $F$, define $\PiBC_{w,d}(F)$ to the the number
% of path families of type $w$ covering $F$ which have $d$ defects.

%\ssec{Type-$\msfBC$ planar networks}

%($\star$ Define type-$\msfBC$ star networks, including the special case of wiring diagrams.)

% %Theorems \ref{t:znetbcpavoid}
% and \ref{t:dnetbcpavoid} suggest defining
% %the following
% type-$\msfBC$ analogs of path families and graphical representation
% %of the sets $\Pi(F)$ and
% (\ref{eq:Gtozsn}), (\ref{eq:Gtozhnq}).

Following \cite{BWHex}, \cite{CSkanTNNChar}, we
%($\star$ Introduce the need to 
consider families
$\pi=(\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n)$
of paths {\em covering} a star network $F \in \net{BC}n$, i.e., using all edges in $F$.
%, define them and their type in $\bn$.)
%We now define type-$\msfBC$ path families and graphical representation of the sets $\Pi(F)$ and (\ref{eq:Gtozsnbc}), (\ref{eq:Gtozbnq}). 
%Given $F \in \net{BC}n$,
%and
%path family
%$\pi=(\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n)$
%covering $F$, 
Call $\pi$ a {\em $\msfBC$-path family} if
for each factor $\smash{F_{[a,b]}}$ of $F$ %$F'_{[a,b]}$ of $F$ %Replacing F'_[a,b] with F_[a,b] to match notation from main result
and each index $i \in [1,n]$, there exist indices $j$, $k$, such that
paths $\pi_i$ and $\pi_{\ol i}$ enter $F_{[a,b]}$ via sources $j$, $\ol j$
and exit $F_{[a,b]}$ via sinks $k$, $\ol k$, respectively.
%if for $i = \ol n, \dotsc, n$ 
%In other words, 
That is, $\pi_{\ol i}$ must be the reflection of $\pi_i$.
Define $\pi$ to have {\em type $u = u_{\ol n} \cdots u_{\ol 1} u_1 \cdots u_n \in \bn$} if for all $i$, path $\pi_i$ begins at source $i$ and terminates at sink $u_i$.
For $F \in \net{BC}n$ and $u \in \bn$, define the sets
\begin{equation}\label{eq:BCpathfams}
  \begin{gathered}
    \PiBC(F) = \{ \pi \,|\, \pi \text{ a $\msfBC$-path family covering } F \},\\
    \PiBC_u(F) = \{ \pi \in \PiBC(F) \,|\, \type(\pi) = u \}.
    \end{gathered}
\end{equation}
%($\star$ Are BC definitions for $\Pi$ needed? Could instead just state that each path $\pi$ is $\msfBC$ iff $F$ is $\msfBC$, for example) $\star$ Yes: see below.\\
For example, 
%consider 
%three path families covering the star network $F$ (\ref{eq:starnetF}),
the 
%type-$\msfBC$ star 
star network 
$F = F_{[1,3]} \circ F_{[2,3]} \circ F_{[1,2]} \circ F_{[\ol1,1]} \in \net{BC}3$
%$F = F_{[1,2]} \circ F_{[2,3]} \circ F_{[\ol 1,1]} 
%\in \net{BC}3$ 
and 
%the following 
path families
%are $\msfBC$-path families satisfying
% $\PiBC(F)$, ($\star$ Make figure for $F$ and fix figures for $\pi$ and $\sigma$ so that $F_{[2,3]}$ vertices and $F_{[\ol1,1]}$ vertex are not vertically aligned. Replacing faux colors with real colors would be fine.)
%in (\ref{eq:pisigma}) 
%%% COLOR LIST FOR REFERENCE: https://www.overleaf.com/learn/latex/Using_colors_in_LaTeX
\begin{equation}\label{eq:pathfamex}
F = \ntnsp
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
  \draw[-, very thick]
  (-.5,2.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick]
  (-.5,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick]
  (-.5,0.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick]
  (-.5,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick]
  (-.5,-2.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  %(-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (2.5,2.5);
  %\draw[-, very thick]
  %(-.5,1.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,1.5) -- (2.5,1.5);
  %\draw[-, very thick]
  %(-.5,0.5) -- (0,1) -- (.5,0.5) -- (1.5,0.5) -- (2.5,-0.5);
  %\draw[-, very thick]
  %(-.5,-0.5) -- (0,-1) -- (.5,-0.5) -- (1.5,-0.5) -- (2.5,0.5);
  %\draw[-, very thick]
  %(-.5,-1.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-1.5) -- (2.5,-1.5);
  %\draw[-, very thick]
  %(-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (2.5,-2.5);
\end{tikzpicture}\ntnsp,
\quad 
%\label{eq:pisigma}
%\nTksp \nTksp \nTksp 
%F =
%\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
%  \node at (-.9,2.5) {$\scriptstyle 3$};
%  \node at (-.9,1.5) {$\scriptstyle 2$};
%  \node at (-.9,0.5) {$\scriptstyle 1$};
%  \node at (3.9,2.5) {$\scriptstyle 3$};
%  \node at (3.9,1.5) {$\scriptstyle 2$};
%  \node at (3.9,0.5) {$\scriptstyle 1$};
%  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
%  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
%  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
%  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
%  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
%  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
%  \draw[-, very thick]
%  (-.5,2.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
%  \draw[-, very thick]
%  (-.5,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
%  \draw[-, very thick]
%  (-.5,0.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (3,0) -- (3.5,0.5);
%  \draw[-, very thick]
%  (-.5,-0.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (3,0) -- (3.5,-0.5);
%  \draw[-, very thick]
%  (-.5,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
%  \draw[-, very thick]
%  (-.5,-2.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
%  %(-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (2.5,2.5);
  %\draw[-, very thick]
  %(-.5,1.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,1.5) -- (2.5,1.5);
  %\draw[-, very thick]
  %(-.5,0.5) -- (0,1) -- (.5,0.5) -- (1.5,0.5) -- (2.5,-0.5);
  %\draw[-, very thick]
  %(-.5,-0.5) -- (0,-1) -- (.5,-0.5) -- (1.5,-0.5) -- (2.5,0.5);
  %\draw[-, very thick]
  %(-.5,-1.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-1.5) -- (2.5,-1.5);
  %\draw[-, very thick]
  %(-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (2.5,-2.5);
%\end{tikzpicture},
%\quad
  \pi = 
% \begin{tikzpicture}[scale=.4,baseline=0]%s1s0s2
%   \node at (-.9,2.5) {$\scriptstyle 3$};
%   \node at (-.9,1.5) {$\scriptstyle 2$};
%   \node at (-.9,0.5) {$\scriptstyle 1$};
%   \node at (1.9,2.5) {$\scriptstyle 3$};
%   \node at (1.9,1.5) {$\scriptstyle 2$};
%   \node at (1.9,0.5) {$\scriptstyle 1$};
%   \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
%   \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
%   \node at (1.9,-2.5) {$\scriptstyle {\ol 3}$};
%   \node at (1.9,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (1.9,-0.5) {$\scriptstyle {\ol 1}$};
%   \draw[-, very thick]
%   (-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5);
%   \draw[-, dotted, very thick]
%   (-.5,1.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,1.5);
%   \draw[-, dashed, very thick]
%   (-.5,0.5) -- (0,1) -- (.5,0.5) -- (1,0) -- (1.5,0.5);
%   \draw[-, thin]
%   (-.5,-0.43) -- (0,-.93) -- (.5,-0.43) -- (1,0.07) -- (1.5,-0.43);
%   \draw[-, thin]
%   (-.5,-0.57) -- (0,-1.07) -- (.5,-0.57) -- (1,-0.07) -- (1.5,-0.57);
%   \draw[-, dotted, thick]
%   (-.5,-1.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-1.5);
%   \draw[-]
%   (-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5);
% \end{tikzpicture}
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
  \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  %\draw[-, color=magenta, very thick]
  %(-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (2.5,2.5);
  %\draw[-, color=green, very thick]
  %(-.5,1.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,1.5) -- (2.5,1.5);
  %\draw[-, color=cyan, very thick]
  %(-.5,0.5) -- (0,1) -- (.5,0.5) -- (1.5,0.5) -- (2.0,0) -- (2.5,0.5);
  %\draw[-, color=purple, very thick]
  %(-.5,-0.5) -- (0,-1) -- (.5,-0.5) -- (1.5,-0.5) -- (2.0,0) -- (2.5,-0.5);
  %\draw[-, color=brown, very thick]
  %(-.5,-1.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-1.5) -- (2.5,-1.5);
  %\draw[-, color=blue, very thick]
  %(-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (2.5,-2.5);
\end{tikzpicture}\ntnsp,
  \quad 
  \sigma = \ntnsp
%   \begin{tikzpicture}[scale=.4,baseline=0]%s1s0s2
%   \node at (-.9,2.5) {$\scriptstyle 3$};
%   \node at (-.9,1.5) {$\scriptstyle 2$};
%   \node at (-.9,0.5) {$\scriptstyle 1$};
%   \node at (1.9,2.5) {$\scriptstyle 3$};
%   \node at (1.9,1.5) {$\scriptstyle 2$};
%   \node at (1.9,0.5) {$\scriptstyle 1$};
%   \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
%   \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
%   \node at (1.9,-2.5) {$\scriptstyle {\ol 3}$};
%   \node at (1.9,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (1.9,-0.5) {$\scriptstyle {\ol 1}$};
%   \draw[-, very thick]
%   (-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,1.5);
%   \draw[-, dotted, very thick]
%   (-.5,0.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,2.5);
%   \draw[-, dashed, very thick]
%   (-.5,1.5) -- (0,1) -- (.5,0.5) -- (1,0) -- (1.5,-0.5);
%   \draw[-]
%   (-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-1.5);
%   \draw[-, dotted, thick]
%   (-.5,-0.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-2.5);
%   \draw[-, thin]
%   (-.5,-1.43) -- (0,-.93) -- (.5,-0.43) -- (1,0.07) -- (1.5,0.57);
%   \draw[-, thin]
%   (-.5,-1.57) -- (0,-1.07) -- (.5,-0.57) -- (1,-.07) -- (1.5,0.43);
% \end{tikzpicture}
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
  \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1)  -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  %\draw[-, color=magenta, very thick]
  %(-.5,2.5) -- (.5,2.5) -- (1,2) -- (1.5,1.5) -- (2.5,1.5);
  %\draw[-, color=green, very thick]
  %(-.5,1.5) -- (0,1) -- (.5,0.5) -- (1.5,0.5) -- (2.0,0) -- (2.5,-0.5);
  %\draw[-, color=cyan, very thick]
  %(-.5,0.5) -- (0,1) -- (.5,1.5) -- (1,2) -- (1.5,2.5) -- (2.5,2.5);
  %\draw[-, color=purple, very thick]
  %(-.5,-0.5) -- (0,-1) -- (.5,-1.5) -- (1,-2) -- (1.5,-2.5) -- (2.5,-2.5);
  %\draw[-, color=brown, very thick]
  %(-.5,-1.5) -- (0,-1) -- (.5,-0.5) -- (1.5,-0.5) -- (2.0,0) -- (2.5,0.5);
  %\draw[-, color=blue, very thick]
  %(-.5,-2.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-1.5) -- (2.5,-1.5);
\end{tikzpicture}\ntnsp, \quad
  \tau = \ntnsp
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-0.9,2.5) {$\scriptstyle 3$};
  \node at (-0.9,1.5) {$\scriptstyle 2$};
  \node at (-0.9,0.5) {$\scriptstyle 1$};
   \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle \ol 3$};
  \node at (-0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-0.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle \ol 3$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
   \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  %\draw[-, color=blue, very thick]
  %(-1,1.5) -- (0,.5) -- (1,1.5);
  %\draw[-, color=red, very thick]
  %(-1,0.5) -- (0,1.5) -- (1,0.5);
  %\draw[-, color=red, ultra thick, densely dashed]
  %(-1,-0.5) -- (-.5,-1) -- (0,-.5) -- (.5,-1) -- (1,-0.5);
  %\draw[-, color=blue, ultra thick, densely dashed]
  %(-1,-1.5) -- (-.5,-1) -- (0,-1.5) -- (.5,-1) -- (1,-1.5);
\end{tikzpicture}\ntnsp,
\end{equation}
%with
%belong to
%The first two path families
satisfy 
$\pi \in 
%\PiBC_e(F_{3\ol12})$
\PiBC_{123}(F)$,
%and
%, 
$\sigma \in 
\PiBC_{3\ol12}(F)$,
$\tau \not \in \PiBC_{123}(F)$, with
%but
%\PiBC_{3\ol12}(F)$, respectively.
%On the other hand,
%$\pi
%while
% the path family 
% \begin{equation}\label{eq:notBCpathfam}
%   \tau = 
% \begin{tikzpicture}[scale=.4,baseline=0]%s1s0s2
%   \node at (-1.4,1.5) {$\scriptstyle 2$};
%   \node at (-1.4,0.5) {$\scriptstyle 1$};
%   \node at (1.4,1.5) {$\scriptstyle 2$};
%   \node at (1.4,0.5) {$\scriptstyle 1$};
%   \node at (-1.4,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (-1.4,-0.5) {$\scriptstyle {\ol 1}$};
%   \node at (1.4,-1.5) {$\scriptstyle {\ol 2}$};
%   \node at (1.4,-0.5) {$\scriptstyle {\ol 1}$};
%   \draw[-, very thick]
%   (-1,1.5) -- (0,.5) -- (1,1.5);
%   \draw[-, dotted, very thick]
%   (-1,0.5) -- (0,1.5) -- (1,0.5);
%   \draw[-, dotted, thick ]
%   (-1,-0.5) -- (-.5,-1) -- (0,-.5) -- (.5,-1) -- (1,-0.5);
%   \draw[-, thick]
%   (-1,-1.5) -- (-.5,-1) -- (0,-1.5) -- (.5,-1) -- (1,-1.5);
% \end{tikzpicture}
% \end{equation}
$\tau$
failing to be a
%is not a 
$\msfBC$-path family
%does not belong to $\PiBC(F_{21} \circ F_{21})$,
%even though it has type $e \in \mfb2$.
because $\tau_1, \tau_{\ol 1}$ in blue (also $\tau_2, \tau_{\ol 2}$ in red) are not reflections of one another.

Each element $F \in \net{BC}n$ combinatorially interprets a product of Kazhdan--Lusztig basis elements in $\hbnq$.
% For $F 
% %= F_{[c_1,d_1]} \circ \cdots \circ F_{[c_t,d_t]}$ 
% \in \net{BC}n$,
% the set $\PiBC(F)$ associates an element of 
% %$\zbn$ and 
% $\hbnq$
% to $F$. 
% %Specifically, we say that
% %$F$ {\em graphically represents}
% %\begin{equation}\label{eq:Gtozsnbc}
% %\sum_{\pi \in \PiBC(F)} \nTksp %\type(\pi)
% %\end{equation}
% %{\em as an element of $\zbn$}.
To describe this interpretation,
%element,
%e corresponding element of $\hbnq$ 
we 
%first 
extend
the 
%definition of 
defect statistic~\cite{BWHex},
\cite{CSkanTNNChar},
\cite{Deodhar90}
%in type-$\msfA$ 
%introduced 
described in \Cref{sec: introduction}.
% {\em type-$\msfA$ defects} from
% \Cref{ssec: type A} 
% Type-$\msfA$ planar networks
%or all $F \in \net A{[h,l]}$,
%the set $\Pi(F)$ also associates an element of
%$H_{[h,l]}(q)$ to $F$.
%To describe this element explicitly, we first assume
%Assume that $F$
%%$F \in \snet{BC}{[\ol n,n]}$
%is a
%%formed by some iteration of 
%%ordinary or condensed 
%concatenation
%of simple star networks 
%%Define $t$ internal vertices $\vertex %z_1, \dotsc, \vertex z_t$
%Each factor $F_{[c_k, d_k]}$ %contributes a single internal vertex
%if $c_k = \ol{d_k}$, and two such %vertices otherwise.
%%Observe that
%%the intersection of two source-to-%sink paths $\PiBC_i$, $\PiBC_j$ in $F$
%must be a disjoint union of the above internal vertices of $F$
%and paths between these.
Given a $\msfBC$-path family $\pi$ covering $F$,
define a {\em type-$\msfBC$ defect} of $\pi$ to be
a triple $(\pi_i, \pi_j, k)$ with
%\item $i \neq j$,
%$(1)$ 
$|i| \leq j$, and
%\item $0 < j$,
%$(2)$ 
$\pi_i$, $\pi_j$ meeting at one of the internal vertices of
  $F_{[c_k,d_k]}$ after having crossed an odd number of times.
Let $\dfctbc(\pi)$ denote the number of type-$\msfBC$ defects of $\pi$.
%(We follow the definition of \cite{CSkanTNNChar}, See \cite{BWHex}, \cite{Deodhar90}.)
%($\star$ Cite Deodhar90, Billey--Warrington.)
Observe that a single vertex can be the location of more than one defect.
For example, consider the star network
%$F'_{[\ol2,2]} \circ F'_{[\ol1,1]} \circ F'_{[1,2]} \circ F'_{[\ol2,2]}
%\in \net {BC}{[\ol2,2]}$
%($\star$ change this, because concatenation $\circ$
%is not included in star network definition?)
and $\msfBC$-path family
%$F_{[1,3]} \circ F_{[1,2]} \circ F_{[1,3]}$
%in $\snet A{[1,3]}$ and
%path family $\pi = (\PiBC_1,\PiBC_2,\PiBC_3,\PiBC_4)$,
\begin{equation}\label{eq:all010}
  F_{[\ol2,2]} \circ F_{[\ol1,1]} \circ F_{[1,2]} \circ F_{[\ol2,2]} =
\begin{tikzpicture}[scale=.5,baseline=-5]
\node at (-.4,1.5) {$\scriptstyle{2}$};
\node at (-.4,0.5) {$\scriptstyle{1}$};  
\node at (-.4,-0.5) {$\scriptstyle{\ol1}$};
\node at (-.4,-1.5) {$\scriptstyle{\ol2}$};  
\node at (4.4,1.5) {$\scriptstyle{2}$};
\node at (4.4,0.5) {$\scriptstyle{1}$};  
\node at (4.4,-0.5) {$\scriptstyle{\ol1}$};
\node at (4.4,-1.5) {$\scriptstyle{\ol2}$};  
\draw[-, very thick] (0,1.5) -- (1,-1.5) -- (2,-1.5) -- (3,-.5) -- (3.5,0) -- (4,.5);
\draw[-, very thick] (0,-1.5) -- (1,1.5) -- (2,1.5) -- (3,.5) -- (3.5,0) -- (4,-.5);
\draw[-, very thick] (0,.5) -- (1,-.5) -- (2,.5) -- (3,1.5) -- (3.5,0) -- (4,1.5);
\draw[-, very thick] (0,-.5) -- (1,.5) -- (2,-.5) -- (3,-1.5) -- (3.5,0) -- (4,-1.5);
\end{tikzpicture},\qquad
\pi = 
\begin{tikzpicture}[scale=.5,baseline=-5]
\node at (-.4,1.5) {$\scriptstyle{2}$};
\node at (-.4,0.5) {$\scriptstyle{1}$};  
\node at (-.4,-0.5) {$\scriptstyle{\ol1}$};
\node at (-.4,-1.5) {$\scriptstyle{\ol2}$};  
\node at (4.4,1.5) {$\scriptstyle{2}$};
\node at (4.4,0.5) {$\scriptstyle{1}$};  
\node at (4.4,-0.5) {$\scriptstyle{\ol1}$};
\node at (4.4,-1.5) {$\scriptstyle{\ol2}$};  
\draw[-, very thick, red]
(0,1.5) -- (1,-1.5) -- (2,-1.5) -- (3,-.5) -- (3.5,0) -- (4,-.5);
\draw[-, very thick, blue]
(0,.5) -- (1,-.5) -- (2,.5) -- (3,1.5) -- (3.5,0) -- (4,1.5);
\draw[-, very thick, densely dashed, blue]
(0,-.5) -- (1,.5) -- (2,-.5) -- (3,-1.5) -- (3.5,0) -- (4,-1.5);
\draw[-, very thick, densely dashed, red]
(0,-1.5) -- (1,1.5) -- (2,1.5) -- (3,.5) -- (3.5,0) -- (4,.5);
\end{tikzpicture}.
\end{equation}
We have $\dfct^{\msfBC}(\pi) = 4$: 
%Type-$\msfBC$ 
defects of $\pi$ are
%the four triples 
$(\pi_{\ol1}, \pi_1, 2)$,
$(\pi_{\ol1}, \pi_2, 3)$,
$(\pi_1, \pi_2, 4)$,
%$(\PiBC_{\ol1}, \PiBC_2, 4)$,
$(\pi_{\ol2}, \pi_2, 4)$.
%and we 
%In general we say that $F$ {\em graphically represents}
%\begin{equation}\label{eq:Gtozbnq}
%\sum_{\pi \in \PiBC(F)} \nTksp %q^{\dfct^{\msfBC}(\pi)} T_{\type(\pi)}
%\end{equation}
%{\em as an element of $\hbnq$}.
%($\star$ Make sure the above is correct.)
%Specializing at $q=1$, we see that if $F$ graphically represents
%$D(q)$ as an element of $\hbnq$, then
%it graphically represents
%$D(1)$ as an element of $\zbn$.
%($\star$ Write a type-$\msfBC$ analog of the following sentence.)
%Given a planar network $F$, define $\PiBC_{w,d}(F)$ to be the the number
%of path families of type $w$ covering $F$ which have $d$ defects.
Refining the sets
%We will find it convenient to partition the sets
%Define an additional set, extending the concept in 
%defined in
(\ref{eq:BCpathfams})
%into smaller sets 
by counting defects of path families in them, 
%we have
let us define
%, to specify defects of a path family:
\begin{equation}\label{eq:BCpathfams_type_defect}
\PiBC_{u,d}(F) = \{ \pi \in \PiBC_u(F) \,|\, \dfctbc(\pi) = d \}.
\end{equation}



%\ssec{Type $\msfBC$}
\section{Main results}\label{s:main}

% $\star$ Think of a plain English introduction to this result. 

When one compares concatenations $F'$ of $k-1$ simple star networks and $F$ of $k$ simple star networks, one sees bijections between certain sets of path families in $F$ and $F'$.
% We now introduce a bijection result that will be useful in the induction proof of our main result. 
%($\star$ Need to define $J_{[a_k,b_k]}$ at some point before Proposition \ref{p:BCbijectiontoshorterpathreversals})  ($\star$ Or we can just call it $J$ in the proposition, and if another $J_{[c,d]}$ must be mentioned, we can call that one $I$, $K$, or $L$.)

\begin{prop}\label{p:BCbijectiontoshorterpathreversals}
  Fix a sequence $(s_{[a_1,b_1]}, \dotsc, s_{[a_{k},b_k]})$
  of reversals in $\bn$, define the generator subset $J 
  = J_{[a_k,b_k]}$,
  %= \{s_{a_k}, \dotsc, s_{b_k-1}\}$,
  and choose an element $w \in \Wejm$.
  %satisfying $wv > w$ for all $v \leq s_{[a_k,b_k]}$.
  Consider two type-$\msfBC$ star networks
  \begin{equation*}
    F' = F_{[a_1,b_1]} \circ \cdots \circ F_{[a_{k-1}, b_{k-1}]}
    \qquad\text{ and }\qquad
    F = F' \circ F_{[a_k,b_k]}.
%    F = F_{[a_1,b_1]} \circ \cdots \circ F_{[a_{k}, b_{k}]}.
  \end{equation*}
  Then for each element $v \in %\mfb{[a_k,b_k]}$ ($\star$ or) 
  W_J$ and each $d \geq 0$, we have a bijection
\begin{equation}\label{eq:BCbijectionsreversals}
%    \PiBC_{w,d}(F)  \quad \overset{1-1}{\longleftrightarrow}\quad  
\PiBC_{wv,d}(F)
    \quad \overset{1-1}{\longleftrightarrow} \ \
    \bigcup_{u \in W_J} %\mfb{[a_k,b_k]}}
    %\ntksp
    \PiBC_{wu\ntnsp,\; d-\ell(u)}(F').
\end{equation}
%($\star$ Would things be simpler if we reversed the roles of $u^{-1}$ and $u$ in the proposition and/or proof?)
%($\star$ Should we be using $\PiBC_{w,d}(F)$ instead of $\PiBC_{w,d}(F)$, or is type-BC clear from context?)
%($\star$ Add condition on $d$ to avoid negative defect, $d \geq \ell(u)$)
  \end{prop}
  % ($\star$ Continue polishing this proof.)
  \begin{proof}
  First we demonstrate the bijection 
  %between the first and third sets in 
  (\ref{eq:BCbijectionsreversals})
  in the case that $v = e$.
% ($\star$ We now note that the paths of $\pi$ ending in positions $\mathrm{max}(1,a_k)$ to $b_k$ are in order by \Cref{l:minrepprop} and thus by \emph{skew symmetric}, paths of $\pi$ ending in positions $\mathrm{min}(\ol{1}, \ol{a}_k)$ to $\ol{b}_k$ are also in order.)
%  For each element $u \in W_J$, let $\sigma_u$ be the unique path family of type $u$ covering $F_{[a_k,b_k]}$.
  Observe that each path family $\pi$ in $\PiBC_w(F)$
  % has the form
  can be decomposed uniquely as the concatenation of its truncation $\pi'$ to $F'$ with its truncation $\pi''$ to $F_{[a_k,b_k]}$: $\pi = \pi' \circ \pi''$.  These truncations
  satisfy $\type(\pi')\type(\pi'') = w$ and $\type(\pi'') \in W_J$.  Thus we have $\pi' \in \PiBC_{wu}(F')$ for some $u \in W_J$.  Let us write the first truncation map $\pi \mapsto \pi'$ as 
  % \begin{equation}\label{eq:trunc}
  % \psi: \PiBC_w(F) \rightarrow \bigcup_{u \in W_J} \PiBC_{wu}(F').
  % \end{equation}
%   \begin{equation}\label{eq:kappadef}
%  \begin{aligned}
    $\trunc: \Pi(F) \rightarrow \Pi(F')$.
%    \pi &\mapsto \text{truncation to $F'$}.
%    \end{aligned}
%    \end{equation}
    Let $\psi$ be the restriction of $\trunc$ to $\PiBC_w(F)$. We claim that the map    
    \begin{equation}\label{eq:psidef}
    \psi:\PiBC_w(F) \rightarrow \bigcup_{u \in W_J} \PiBC_{wu}(F')
    \end{equation}
  is bijective.  
  To see this, keep $w \in \Wejm$ fixed, and choose a
  path family $\tau$ belonging to
  %element $u \in W_J$ with 
  $\cup_{u \in W_J} \PiBC_{wu}(F')$ and define $y = y(\tau) = w^{-1}\type(\tau) \in W_J$. %be the element of $W_J$ such that $\type(\tau) = wy$.
  %for some $u = u(\tau) \in W_J$.
  %\neq \emptyset$
  %and a path family $\tau \in \PiBC_{wu}(F')$.  
  The set $\PiBC_{y^{-1}}(F_{[a_k,b_k]})$ contains a unique element, call it $\sigma_{y^{-1}}$.
  %Now define a 
  It is easy to see that the map 
  \begin{equation}\label{eq:kappadef}
  \begin{aligned}
    \kappa: {\bigcup_{u \in W_J}} \PiBC_{wu}(F') &\rightarrow \PiBC_w(F)\\
    \tau &\mapsto \tau \circ \sigma_{y(\tau)^{-1}}
    \end{aligned}
    \end{equation}
     %It is easy to see that $\kappa$ 
     inverts $\psi$. %$\PiBC_w(F)$ and satisfies $\psi(\tau \circ \sigma_{u^{-1}}) = \tau$.
%\begin{equation}\label{eq:restriction}
%  \pi = \pi' \circ \sigma_{u^{-1}}, 
%\end{equation}
%where $\pi'$ is the restriction of $\pi$ to $F'$, 
% where $u \in W_J$ and $\pi' \in \PiBC_{wu}(F')$.
% First version of the following sentence:
% Conversely, for every element $u \in W_J$ and every path family $\pi' \in \PiBC_{wu}(F')$ we have that $\pi' \circ \sigma_{u^{-1}}$ belongs to $\PiBC_{w}(F)$.  Thus the map 
% \begin{equation}
% \psi: \PiBC_w(F) \rightarrow 
 %\bigcup_{u \in W_J} \PiBC_{wu}(F')
%\{ (\pi',u) \in \Pi(F') \times W_J \,|\, \type(\pi') = wu \}
%\end{equation}
%defined by (\ref{eq:restriction})
%is bijective.  
Now we claim that for $\pi \in \PiBC_{w}(F)$, the paths $\pi$, $\pi' = \psi(\pi)$ satisfy
%decomposition (\ref{eq:restriction}) satisfies
%if $\psi(\pi) = (\pi',u)$, then 
\begin{equation*}\dfctbc(\pi') = \dfctbc(\pi) - \ell(u).
\end{equation*}
To see this, observe that \Cref{l:minrepprop} 
%($\star$ or the definition of $\Wejm$?) 
implies 
% ($\star$ What is the point of this paragraph?  It looks like we are demonstrating that some map is bijective, but we haven't defined the map.)
% %% Define the map 
% % $\pi' \mapsto \pi' \concat \sigma_{u}$
% First we set $v=e$ on the left-hand side of (\ref{eq:BCbijectionsreversals}). Now we consider all possible path family pairs $\pi'$ and $\sigma_{u^{-1}}$ such that $\pi' \circ  \sigma_{u^{-1}} = \pi$. 
% It is obvious that the map $\pi' \mapsto \pi' \circ \sigma_{u^{-1}} = \pi$ is injective, as the type of $\pi'$ can always be uniquely written as $wu$, for $w \in \Wejm$ and $u^{-1} \in W_J$. 
% Further, note that if $\pi$ is of type $w$, this directly implies that that $\pi'$ must be of type $wu$ if $\sigma_{u^{-1}}$ is of type $u^{-1}$, and thus the mapping is also surjective, which completes the bijectivity of the map. %Bijectivity of the mapping
% leftmost and rightmost sides are in bijection
%We now note that 
the paths of $\pi = \pi' \circ \sigma_{u^{-1}}$ terminating at certain sinks %$\mathrm{max}\{1,a_k\}, \dotsc, b_k$ of $F$ 
% and $\ol{a_k}$ to $\ol{b_k}$
to have increasing indices: %increasing how? maybe "ordered" instead?
%Specifically, these are 
sinks
$[\ol b_k, b_k]$
if $a_k = \ol{b_k}$,
and sinks $[\ol{b_k}, \ol{a_k}]$ and $[a_k,b_k]$ if $0 < a_k < b_k$.
%, while those terminating at sinks
%.
%are, by definition, in order, as seen in . %type(pi) = w
%Further by \Cref{l:minrepprop}, 
%Similarly, paths from 
%$\mathrm{min}\{\ol{1}, \ol{a_k}\}, \dotsc, \ol{b_k}$ of $F$ have decreasing indices. 
Therefore by \Cref{l:longinversions},
%($\star$ and/or another fact?), 
the paths of $\pi'$  terminating at the corresponding sinks %$\mathrm{max}\{1, a_k\}, \dotsc, b_k$ and $\mathrm{min}\{\ol{1}, \ol{a_k}\}, \dotsc, \ol{b_k}$ 
of $F'$ include $\ell(u^{-1}) = \ell(u)$ path pairs $(\pi'_i, \pi'_j)$ with $|i| \leq j$ and with $\pi'_j$ terminating
at a sink with smaller 
%(positive?) 
index than that of $\pi'_i$.  
%($\star$ Is this correct?) 
% (TJP - I agree the above is correct)
%, out of order.
%This implies that $\sigma_{u^{-1}}$ puts each pair ($\pi_i$, $\pi_j$) back in order, as $\pi' \circ \sigma_{u^{-1}}$ is of type $w$.  
%Could write: %Thus $\ell(u)$ pairs of path of pi' ending in positions a_k to b_k in F' are out of order. 
Each such pair has crossed an odd number of times in $F'$ and crosses again, defectively, in $F_{[a_k,b_k]}$.
Thus $\dfctbc(\pi') = 
%and therefore if $\pi$ has $d$ defects, then $\pi'$ has 
\dfctbc(\pi) - \ell(u)$ 
%defects 
and the bijection (\ref{eq:psidef}) restricts for all $d$ to
%Finally, if we consider all possible $u \in W_J$, the mapping $\pi' \mapsto \pi' \circ \sigma_u = \pi$ adds $\ell(u)$ defects, and thus 
\begin{equation*}
      \PiBC_{w,d}(F)  \overset{1-1}{\longleftrightarrow}\bigcup_{u \in W_J}
    \PiBC_{wu,d-\ell(u)}(F').
\end{equation*}


We complete the proof of the bijections (\ref{eq:BCbijectionsreversals}) by
demonstrating bijections between all pairs of sets $\PiBC_{wv,d}(F)$ and  $\PiBC_{wy,d}(F)$ with $v, y \in W_J$. %different from $e$.
%the first and the second sets in~\eqref{eq:BCbijectionsreversals}.
% Choose $v\in W_J$ and define a map $\phi_v: \PiBC_{w,d}(F) \rightarrow  \PiBC_{wv,d}(F)$ 
In particular, for fixed $v, y \in W_J$, %For such an element $v$ and an arbitrary element $y$ in $W_J$,
%Choose $v\in W_J$ and 
define the map 
%($\star$ is this correct?)
%\begin{equation}
%\begin{aligned}
$\phi_{v,y}: \PiBC_{wv}(F) \rightarrow  \PiBC_{wy}(F)$ by writing
$\pi = \trunc(\pi) \circ \sigma_{z^{-1}}$
for some $z \in W_J$ and by mapping
\begin{equation}\label{eq:phivy}
\pi \mapsto \mathrm{trunc}(\pi) \circ \sigma_{z^{-1}v^{-1}y}.
\end{equation}
To see that $\phi_{v,y}$ is well defined, observe that since the type of $\trunc(\pi)$ is $wvz$, the type of $\phi_{v,y}(\pi)$ is $wvz z^{-1}v^{-1}y = wy$.  To see that $\phi_{v,y}$ is invertible, note that 
$\phi_{y,v}(\phi_{v,y}(\pi))$ equals
%its inverse is $\phi_{y,v}$:
\begin{equation*}
%\begin{aligned}
\phi_{y,v}(\trunc(\pi) \circ \sigma_{z^{-1}v^{-1}y})
    = \trunc(\pi) \circ \sigma_{(z^{-1}v^{-1}y) y^{-1}v}\\
    = \trunc(\pi) \circ \sigma_{z^{-1}}
    = \pi.
%    \end{aligned}
\end{equation*}
%$\pi=\mathrm{trunc}(\pi)\circ \sigma_{z^{-1}}\mapsto \mathrm{trunc}(\pi)\circ\sigma_{z^{-1}v^{-1}y}$
%Let $\tau = \mathrm{trunc}(\pi)\circ \sigma_{z^{-1}v^{-1}y}$
%Then $\phi_{y,v}(\tau)=\mathrm{trunc}(\pi)\circ\sigma_{z^{-1}v^{-1}yy^{-1}v}=\mathrm{trunc}(\pi)\circ\sigma_{z^{-1}} $
%$\phi_{y,v}(\tau)=\mathrm{trunc}(\tau)\circ \sigma_{\type(\mathrm{trunc}(\tau))^{-1}wv}=\mathrm{trunc}(\pi)$
% ($\star$ what is $u^{-1}$ in the above equation?)
%by decomposing $\pi$ as $\pi = \pi' \circ \sigma_{u^{-1}v}.
%\end{aligned}
%\end{equation}
%with $u^{-1}\in W_J$
%and setting $\phi_v(\pi) = \pi' \circ \sigma_{u^{-1}v}$.
%Clearly this map is bijective with inverse $\phi_{v^{-1}}$.
Finally, we claim that $\phi_{v,y}$ restricts to a bijection
from $\smash{\PiBC_{wv,d}(F)}$ to $\smash{\PiBC_{wy,d}(F)}$ for all $d$.
%Furthermore, $\phi_v$ preserves defects. 
To see this, observe that as $z$ varies over $W_J$, the set of paths of $\trunc(\pi) \circ \sigma_{z^{-1}}$ which meet at the central vertex of $F_{[a_{k},b_{k}]}$ is constant.
% ,
% regardless of whether they cross there. Thus $\dfctbc(\pi' \circ \sigma_{z^{-1}})$ is also constant as $z$ varies over $W_J$.
%, and $\phi_{v,y}$ restricts to a bijection from $\PiBC_{wv,d}(F)$ to $\PiBC_{wy,d}(F)$.
% To see that $\phi_v$ is well-defined, observe that 
% for all elements $y \in W_J$ we have $\dfctbc(\pi' \circ \sigma_y) = \dfctbc(\pi' \circ \sigma_e)$.
% for all elements $v \in W_J$ The concatenation $\pi' \circ \sigma_y$ introduces a fixed number of new defects 
% $\sigma_u$ and $\sigma_{u^{-1}v}$
% $\dfctbc(\pi) = \dfctbc(\phi(\pi))$
% Consider $\pi$ and $\tau$ of types $w$ and $wv$ respectively, where $v \in W_J$. 
% Then let us decompose $\pi$ as $\pi = \pi' \circ \sigma_{u^{-1}}$, with $u^{-1} \in W_J$, and type$(\pi') = wu$. 
% But then, let us define $\phi(\pi) = \pi' \circ \sigma_{u^{-1}v} = \tau \in \PiBC_{wv}(F)$. 
% Since right multiplication by $v$ is a bijection from $W_J$ to $W_J$, $\phi$ is bijective. 
% % Since that final vertex is irrelevant for defects, as all the wires in that final star all cross or touch regardless, 
% %% Rewriting the 'Since that final vertex...' line ------
% The multiplication by $v$ 
% %in $F_{[a_k,b_k]}$
% is irrelevant for defects, because all wires in the final star touch, regardless of crossing or not. 
% Thus
% %% ----- ^ please check or revise this phrasing
% we have $\dfctbc(\pi) = \dfctbc(\phi(\pi)) = \dfctbc(\tau)$. 
% Therefore, 
% \begin{equation*}
%     \PiBC_{w,d}(F)
%     \overset{1-1}{\longleftrightarrow}
%     \PiBC_{wv,d}(F)
% \end{equation*}
% which completes the proof.
  \end{proof}

This leads to our main result: combinatorial formulas for coefficients appearing in the natural expansions of certain products of Kazhdan--Lusztig basis elements. 

% ($\star$ Or some introductory sentence like this.) 

\begin{thm}\label{t:BCwiringdiagramprod}
    Fix a sequence $(s_{[a_1,b_1]}, \dotsc, s_{[a_{k},b_k]})$
  of reversals in $\bn$ and define the type-$\msfBC$ star network
  %wiring diagram
  %\begin{equation}\label{eq:mainF}
    $F = F_{[a_1,b_1]} \circ \cdots \circ F_{[a_{k}, b_{k}]}$.
%  \end{equation}
Then we have
\begin{equation}\label{eq:mainklprodk}
    \wtc{s_{[a_1,b_1]}}q \cdots \wtc{s_{[a_k,b_k]}}q = 
    \sum_{y \in \bn} \sum_{d \geq 0} |\PiBC_{y,d}(F)| q^d T_y.
\end{equation}
\end{thm}
\begin{proof}
(Idea)
% TODO - add base case here
%Observe that f
For any reversal $s_{[a,b]}$ and corresponding generator set $J = J_{[a,b]}$, we have by \Cref{p:wtcreversal} that the coefficient of $T_u$ in $\wtc{s_{[a,b]}}q$ is $1$ for $u \in W_J$ and is $0$ otherwise.
On the other hand, the star network $F_{[a,b]}$ satisfies
\begin{equation*}
    |\PiBC_{u,d}(F_{[a,b]})| = \begin{cases}
        1 &\text{if $u \in W_J$ and $d=0$},\\
        0 &\text{otherwise}.
    \end{cases}
\end{equation*}
Thus the identity (\ref{eq:mainklprodk}) holds for a single Kazhdan--Lusztig basis element indexed by a reversal.
Now assume 
%therefore 
that the identity holds for products of $1, \dotsc, k-1$
such elements and consider a product of $k$ of them. Writing
%\begin{equation*}
%    \begin{cases}
%        1 &\text{if $w \in W_J$},\\
%        0 &\text{otherwise}.
%    \end{cases}
%\end{equation*}
%if $k=1$, then
%\begin{equation*}
%    \wtc {s_{[a,b]}}q = \sum_{w\in W_J} T_w = \sum_{w \in \bn} \sum_{d \geq 0} |\PiBC_{w,d}(F)| q^d T_w$$
%where 
%$|\PiBC_{w,d}| = \begin{cases}
%    1 &\text{if $d=0$ and $w\in \mfb {[a_1,b_1]}$},\\
%    0 &\text{otherwise}.
%  \end{cases}$
%Let 
$F' = F_{[a_1,b_1]} \circ \cdots \circ F_{[a_{k-1}, b_{k-1}]}$, we have
%    Let us assume that formula holds up to $k-1$, as it trivially holds for $k=1$. Thus

\begin{equation}\label{eq:mainklprodkminus1}
        \wtc {s_{[a_1,b_1]}}q \cdots \wtc {s_{[a_{k-1},b_{k-1}]}}q = 
    \sum_{v \in \bn} \sum_{d \geq 0} |\PiBC_{v,d}(F')| q^d T_v.
    \end{equation}
%Right-multiplying both sides of (\ref{eq:mainklprodkminus1}) by
% Now, let us concatenate 
%$\wtc {s_{[a_k,b_k]}}q$, 
Defining $J = J_{[a_k,b_k]}$ and applying \Cref{p:klprodcosetexpansion} to (\ref{eq:mainklprodk}) -- (\ref{eq:mainklprodkminus1})
%and 
we have that the left-hand side of (\ref{eq:mainklprodk}) equals
%\begin{equation*}
%\begin{aligned}
%        \wtc {s_{[a_1,b_1]}}q \cdots \wtc {s_{[a_k,b_k]}}q &= \left(\wtc {s_{[a_1,b_1]}}q \cdots \wtc {s_{[a_{k-1},b_{k-1}]}}q \right) \wtc {s_{[a_k,b_k]}}q \\
%        &= 
%\bigg( \sum_{w \in \bn} \sum_{d \ge 0}|\PiBC_{w,d}(F')| q^d T_w \bigg)  \wtc {s_{[a_k,b_k]}}q. \\
        % &= \sum_{w \in \Wejm} \sum_{v \in  W_{J}} q^{\ell(v)} \left(\sum_{d=0}^{k-1} |\PiBC_{wv,d}(F')| q^d \right) T_w \tilde{C}_{s_{[a_k,b_k]}} .     
%   \end{aligned}       
%\end{equation*}
%By Proposition %\ref{p:klprodcosetexpansion}, 
%letting $J = J_{[a_k,b_k]}$
%and $a_{wv} = \sum_{d\ge0} |\PiBC_{wv,d}(F')| q^d$, we then have
%($\star$ Need to break this up into several sentences.  The notes over equals signs should start new sentences: "By the second bijection in Proposition 5.1", etc.)
\begin{equation}\label{eq:appliedprop34toprod}
%\begin{aligned}
%  \wtc {s_{[a_1,b_1]}}q \cdots \wtc {s_{[a_k,b_k]}}q& =
%&\sum_{w \in \Wejm} \sum_{v \in  W_{J}} \ntksp q^{\ell(v)} \bigg(\sum_{d\ge0} |\PiBC_{wv,d}(F')|  q^d \ntnsp \bigg) T_w \wtc {s_{[a_k,b_k]}}q \\
%=
\sum_{w \in \Wejm} 
\ntksp \bigg( \sum_{u \in  W_{J}} \sum_{d\geq 0} |\PiBC_{wu,d}(F')|  q^{\ell(u)+d}\bigg) T_w \wtc {s_{[a_k,b_k]}}q.
\end{equation}
%Changing the order of summation, 
Applying \Cref{p:BCbijectiontoshorterpathreversals} and using (\ref{eq:TuWJ}),
%the fact that $\PiBC_{wu,d}(F') = \emptyset$ when $d < 0$, 
we may rewrite (\ref{eq:appliedprop34toprod}) as
%By \Cref{p:BCbijectiontoshorterpathreversals}, this equals
\begin{equation}
\begin{aligned}
%\sum_{w \in \Wejm} 
%\ntksp \bigg( \sum_{d\geq 0} q^d %\sum_{u \in  W_{J}}  |\PiBC_{wu,d-\ell(u)}(F')|  \bigg) T_w \wtc {s_{[a_k,b_k]}}q
%&\overset{\text{2nd bij. in Proposition~\ref{p:BCbijectiontoshorterpathreversals}}}{=}
\sum_{w \in \Wejm} 
\ntksp \bigg(  \sum_{d\geq 0} q^d |\PiBC_{w,d}(F)|  \bigg) T_w \wtc {s_{[a_k,b_k]}}q
&=\sum_{w \in \Wejm} 
\ntksp \bigg(  \sum_{d\geq 0} q^d |\PiBC_{w,d}(F)|  \bigg) \sum_{u\in W_J} T_{wu}\\
&=\sum_{w \in \Wejm} \sum_{u\in W_J}
%\ntnsp  
\sum_{d\geq 0} q^d |\PiBC_{w,d}(F)|  T_{wu}\\
%&\overset{\text{1st bij. in Proposition~\ref{p:BCbijectiontoshorterpathreversals}}}{=}
&= \sum_{w \in \Wejm} \sum_{u\in W_J}
%\ntksp  
\sum_{d\geq 0} q^d |\PiBC_{wu,d}(F)|  T_{wu},
%&\overset{\eqref{eq:Jfactor}}{=} \sum_{y\in\bn} \sum_{d\geq 0} q^d |\PiBC_{y,d}(F)|  T_{y}. 
\end{aligned}
\end{equation}
which by (\ref{eq:Jfactor}) equals the right-hand side of (\ref{eq:mainklprodk}).
\end{proof}
%Since for all $v\in \bn$ and $d<0$, we have $|\PiBC_{v,d}(F')|=\emptyset$, we may replace the inner sum by $\sum\limits_{d\geq 0}$ and change the order of summations to obtain ($\star$ to rewrite this last expression as?)
%\begin{equation}\label{eq:sumschanged}
%\begin{gathered}
%\sum_{w \in \Wejm} 
%\ntksp \bigg( \sum_{d\geq 0} q^d \sum_{u \in  W_{J}}  |\PiBC_{wu,d-\ell(u)}(F')| \bigg) T_w \wtc {s_{[a_k,b_k]}}q=\sum_{w \in \Wejm} 
%\ntksp \bigg( \sum_{d\geq 0} q^d   |\PiBC_{wu,d}(F)| \bigg) T_w \wtc {s_{[a_k,b_k]}}q.
%\end{gathered}
%\end{equation}
% Changing the bound of $d$, we may rewrite the parenthesized sum in (\ref{eq:appliedproptoprod}) as
% \begin{equation*}
%     \sum_{u \in  W_{J}} \sum_{d\geq \ell(u)} q^d |\PiBC_{wu,d-\ell(u)}(F')|=\sum_{d\geq \ell(u)}  \sum_{u \in  W_{J}} q^d |\PiBC_{wu,d-\ell(u)}(F')|.
% \end{equation*}
%By the bijection of \Cref{p:BCbijectiontoshorterpathreversals}, we have that $\sum_{u \in  W_{J}} 
%|\PiBC_{wu,d-\ell(u)}(F')| = |\PiBC_{w,d}(F)|$, so Equation ({\ref{eq:sumschanged})
%the previous expression 
%is equivalent to
%\begin{equation*}
%\begin{aligned}
%    \sum_{v \in  W_{J}} q^{\ell(v)} \left(\sum_{d\ge0}|\PiBC_{wv,d}(F')| q^d \right) &= \sum_{v \in W_J} \sum_{d\ge0} q^{\ell(v) + d} |\PiBC_{wv,d}(F')| = \sum_{d\ge0}q^{d} |\PiBC_{w,d}(F)|.
    % = 
% \sum_{d\geq 0} q^d |\PiBC_{w,d}(F)|  = \sum_{d\geq\ell(u)} q^d |\PiBC_{w,d}(F)|+\sum_{d<\ell(u)} q^d |\PiBC_{w,d}(F)|.
    % for $u \in W_J$.
%\end{aligned} 
%\end{equation*}
%($\star$ Explain the above a little more. $u$ is not defined in the above expression)
%Since for any negative integer $c$ we have $|\PiBC_{w,c}(F)| = |\emptyset| = 0$, we can relax the bound of $d$, yielding
%\begin{equation*}
%    \sum_{d\geq 0} q^d |\PiBC_{w,d}(F)|. 
%\end{equation*}
% ($\star$ maybe a changing of variables)
%Replacing the parenthesized sum ($\star$ somewhere in) (\ref{eq:appliedproptoprod}), ($\star$ some expression?) becomes
%\begin{equation*}
%    \sum_{w \in \Wejm} \left(  \sum_{d\ge0}q^{d} |\PiBC_{w,d}(F)|\right)T_w \wtc {s_{[a_k,b_k]}} q
%\end{equation*}
%By 
%the first bijection in 
%Proposition~\ref{p:BCbijectiontoshorterpathreversals} and the bijection~\eqref{eq:Jfactor}, this equals
%we have that (\ref{eq:appliedproptoprod}) equals
%\begin{equation*}
%    \begin{aligned}
%        \wtc {s_{[a_1,b_1]}}q \cdots \wtc {s_{[a_k,b_k]}}q &=  \sum_{w \in \Wejm} \left( \sum_{v \in W_J} \sum_{d\ge0}q^{\ell(v) +d} |\PiBC_{wv,d}(F')| \right) T_w \wtc {s_{[a_k,b_k]}} q \\
%        & = 
%\sum_{w \in \Wejm} \left(  \sum_{d\ge0}q^{d} |\PiBC_{w,d}(F)|\right)T_w \wtc {s_{[a_k,b_k]}} q \\
 %       &
 %= \sum_{v \in \bn} \sum_{d \ge 0} |\PiBC_{v,d}(F)| q^d T_v.
%    \end{aligned}
%\end{equation*}
% ($\star$ Do we need the $\PiBC_{w,d}(F)\overset{1-1}{\longleftrightarrow}\PiBC_{wv,d}(F)$ bijection from \Cref{p:BCbijectiontoshorterpathreversals}? )
% ($\star$ Throughout the paper, we should be consistent with letting $w \in \Wejm$, $u \in W_J$, and $v \in W=\bn$. Or maybe swap $u$ and $v$. (Swapped $u$ and $v$ in Proposition~\ref{p:klprodcosetexpansion}. Updated Corollary~\ref{c:klprodnatexpansion} and Theorem~\ref{t:BCwiringdiagramprod}.) Consider \Cref{l:minrepprop} and the proceeding text, \Cref{p:klprodcosetexpansion}, \Cref{c:klprodnatexpansion}, \Cref{p:BCbijectiontoshorterpathreversals}, and \Cref{t:BCwiringdiagramprod})
%\end{proof}
%($\star$ Include an example?)
%($\star$ Include some open questions?)
\noindent
To illustrate \Cref{t:BCwiringdiagramprod}, consider  %allows us for example to compute 
the coefficient of $T_e$ in $\wtc{s_{[1,3]\ntnsp}}q \wtc{s_{[2,3]\ntnsp}}q \wtc{s_{[1,2]\ntnsp}}q \wtc{s_{[\ol1,1]\ntnsp}}q$.
%Counting defects in p
Exactly four path families of type $e$ cover the network $F = F_{[1,3]} \circ F_{[2,3]} \circ F_{[1,2]} \circ F_{[\ol 1, 1]}$ in (\ref{eq:pathfamex}). These include 
%By 
the defect-free path family 
$\pi$ in (\ref{eq:pathfamex}) and the three path families
\begin{equation*}
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
  \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,2.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5)  -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) -- (.5,1.5) -- (1,2) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-1.5) -- (1,-2) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
%  \draw[fill=none](1,2) circle (0.3) node [black]{};
  \draw[fill=none](2,1) circle (0.3) node [black]{};
\end{tikzpicture},
  \qquad 
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-.9,2.5) {$\scriptstyle 3$};
  \node at (-.9,1.5) {$\scriptstyle 2$};
  \node at (-.9,0.5) {$\scriptstyle 1$};
  \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (-.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle {\ol 3}$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
 \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,1.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (0,1.5) --  (.5,2.5) -- (1,2) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0, -1.5) -- (.5,-1.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  \draw[fill=none](1,2) circle (0.3) node [black]{};
%  \draw[fill=none](2,1) circle (0.3) node [black]{};
\end{tikzpicture}
, \qquad
% \tau = 
\begin{tikzpicture}[scale=.5,baseline=0]%s1s0s2
  \node at (-0.9,2.5) {$\scriptstyle 3$};
  \node at (-0.9,1.5) {$\scriptstyle 2$};
  \node at (-0.9,0.5) {$\scriptstyle 1$};
   \node at (3.9,2.5) {$\scriptstyle 3$};
  \node at (3.9,1.5) {$\scriptstyle 2$};
  \node at (3.9,0.5) {$\scriptstyle 1$};
  \node at (-.9,-2.5) {$\scriptstyle \ol 3$};
  \node at (-0.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (-0.9,-0.5) {$\scriptstyle {\ol 1}$};
  \node at (3.9,-2.5) {$\scriptstyle \ol 3$};
  \node at (3.9,-1.5) {$\scriptstyle {\ol 2}$};
  \node at (3.9,-0.5) {$\scriptstyle {\ol 1}$};
 \draw[-, very thick, green]
  (-.5,2.5) -- (0,1.5) -- (.5,1.5) -- (1,2) -- (1.5,2.5) -- (3.5,2.5);
  \draw[-, very thick, red]
  (-.5,1.5) -- (0,1.5) -- (.5,0.5) -- (1.5,0.5) -- (2,1) -- (2.5,1.5) -- (3.5,1.5);
  \draw[-, very thick, blue]
  (-.5,0.5) -- (0,1.5) --  (.5,2.5) -- (1,2) -- (2,1) -- (3,0) -- (3.5,0.5);
  \draw[-, very thick, densely dashed, blue]
  (-.5,-0.5) -- (0,-1.5) -- (.5,-2.5) -- (1,-2) -- (2,-1) -- (3,0) -- (3.5,-0.5);
  \draw[-, very thick, densely dashed, red]
  (-.5,-1.5) -- (0,-1.5) -- (.5,-0.5) -- (1.5,-0.5) -- (2,-1) -- (2.5,-1.5) -- (3.5,-1.5);
  \draw[-, very thick, densely dashed, green]
  (-.5,-2.5) -- (0, -1.5) -- (.5,-1.5) -- (1,-2) -- (1.5,-2.5) -- (3.5,-2.5);
  \draw[fill=none](1,2) circle (0.3) node [black]{};
  \draw[fill=none](2,1) circle (0.3) node [black]{};
\end{tikzpicture},
\end{equation*}
with circled defects.  The four families contribute $1, q, q, q^2$, respectively, to the coefficient of $T_e$.
%, for a total of
Thus
the desired coefficient
is $1 + 2q + q^2$.

As a corollary of \Cref{t:BCwiringdiagramprod}, one may similarly interpret the natural expansion of products 
$\wtc{w^{(1)}}q \cdots \wtc{w^{(r)}}q \in \hbnq$ when $w^{(1)}, \dotsc, w^{(r)} \in \bn$ \avoidp. By \cite[Theorem\,5.21]{SkanHyperGC}, each
of these Kazhdan--Lusztig basis elements factors as on the left-hand side of (\ref{eq:mainklprodk}), and by the comment preceding \Cref{p:wtcreversal},
each of these elements of $\bn$ corresponds to Schubert varieties which
are simultaneously smooth in types $\msfBB$ and $\msfC$.
It would be interesting to extend the theorem to Hecke algebras of Coxeter groups of type $\mathsf D$.

% \acknowledgements{}

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\end{document}
