%% 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{26}
%% 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{amsfonts}
%\usepackage{relsize}
\usepackage{graphicx}
\usepackage{mathrsfs}
\usepackage{comment}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{enumitem}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{hyperref}
\usepackage{cleveref}
\usepackage[utf8]{inputenc}

%\DeclareTextFontCommand{\emph}{\color{blue}\em}

\newcommand{\GL}{\mathrm{GL}}
\newcommand{\SL}{\mathrm{SL}}
\newcommand{\SO}{\mathrm{SO}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\bs}{\boldsymbol}
%\renewcommand{\S}{\mathfrak{S}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\mc}{\mathcal}
\newcommand{\splt}{\mathsf{split}}
\newcommand{\merge}{\mathsf{merge}}
\renewcommand{\d}{\boldsymbol{d}}
\renewcommand{\tilde}{\widetilde}

\DeclareMathOperator{\T}{\mathcal{T}}
\DeclareMathOperator{\partn}{\mathsf{str}}
\DeclareMathOperator{\toprow}{\mathsf{top}}
\DeclareMathOperator{\colr}{\mathsf{col}}
\DeclareMathOperator{\Fl}{Fl}
\DeclareMathOperator{\gr}{Gr}
\DeclareMathOperator{\coh}{H}

\newcommand\precdot{\mathrel{\ooalign{$\prec$\cr
  \hidewidth\raise0.001ex\hbox{$\cdot\mkern0.6mu$}\cr}}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{def-prop}[theorem]{Definition-Proposition}
\newtheorem{prop}[theorem]{Proposition}
\Crefname{prop}{Proposition}{Propositions}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{introthm}{Theorem}
\renewcommand{\theintrothm}{\Alph{introthm}}

\theoremstyle{definition}
\newtheorem{ex}[theorem]{Example}
\newtheorem{quest}[theorem]{Question}
\newtheorem{defin}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}

\usepackage{lipsum}

%% define your title in the usual way
\title{Bijections between interlacing triangles, Schubert puzzles, and graph colorings}

%% 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{\texorpdfstring{Christian Gaetz\thanks{\href{mailto:gaetz@berkeley.edu}{gaetz@berkeley.edu}}\addressmark{1} \and Yibo Gao\thanks{\href{mailto:gaoyibo@bicmr.pku.edu.cn}{gaoyibo@bicmr.pku.edu.cn}. YG was partially supported by NSFC Grants 12471309.}\addressmark{2}}{Christian Gaetz and Yibo Gao}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, Berkeley, CA, USA. \\ \addressmark{2}Beijing International Center for Mathematical Research, Peking University, Beijing, China.}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\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
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We show that \emph{interlacing triangular arrays}, introduced by Aggarwal--Borodin--Wheeler to study certain probability measures, can be used to compute various kinds of Schubert structure constants. We do this by establishing a splitting lemma, allowing for arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 and certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson--Tao puzzles. This proves one enumerative conjecture of Aggarwal--Borodin--Wheeler and disproves another.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{interlacing triangular array, puzzle, graph coloring, Schubert calculus.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{fpsac.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\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:intro}

The \textit{LLT polynomials} are a $1$-parameter family of symmetric polynomials introduced by Lascoux, Leclerc, and Thibon \cite{LLT}. They have close connections to Macdonald polynomials and Kazhdan--Lusztig theory, among other areas of representation theory and geometry. In recent work, Aggarwal, Borodin, and Wheeler \cite{Aggarwal-Borodin-Wheeler} studied probability measures arising from the Cauchy identity for LLT polynomials. They show that these measures asymptotically split into a continuous part, given by a product of GUE corners processes, and a discrete part, supported on \textit{interlacing triangular arrays}. They conjectured that these arrays are equinumerous with vertex colorings of the triangular grid graph in the "rank-$3$" case, and with vertex colorings of another grid graph in the rank-$4$ case. We construct a bijection between interlacing arrays and vertex colorings, proving the first conjecture, and we disprove the second\footnote{A full version of the work summarized in this extended abstract appears in \cite{arxiv-version}.}.

Our bijection factors through intermediate objects which are certain \textit{edge} colorings of the triangular grid graph. We recognize these edge labelings as cryptomorphic to certain Schubert calculus \emph{puzzles} \cite{Knutson-Tao-equivariant}. We apply various geometric interpretations of puzzles \cite{halacheva-knutson-zinn-justin,Knutson-Zinn-Justin-2,Vakil, wheeler-zinn-justin} to prove that the corresponding families of interlacing triangular arrays compute structure constants in cohomology $H^*(\gr(d,n))$ and $K$-theory $K(\gr(d,n))$ of Grassmannians, in the (localized) cohomology $H^{*\text{loc}}_{\C^{\times}}(T^*\gr(d,n))$ of their cotangent bundles, and for the multiplication in the cohomology of partial flag varieties of classes pulled back from smaller partial flag varieties.

We now define interlacing triangular arrays, the main objects of study.

\begin{defin}[Aggarwal--Borodin--Wheeler \cite{Aggarwal-Borodin-Wheeler}]\label{def:interlacing-triangle}
% ABW uses "rank n (# of triangles) and height N", but it is unthinkable to not work in Sn. We have to go with height n and rank either m or r. 
An \emph{interlacing triangular array} $T$ of rank $m$ and height $n$ is a collection $\{T^{(i)}_{j,k} \mid 1 \leq i \leq m, 1 \leq j \leq k \leq n \}$ of positive integers from $[m]=\{1,2,\ldots,m\}$, subject to the following conditions:
\begin{itemize}
    \item[(a)] For each $k=1,\ldots,n$ we have an equality of multisets:
    \[\{T^{(i)}_{j,k} \mid 1 \leq i \leq m, 1 \leq j \leq k\} = \{1^k\} \cup \cdots \cup \{m^k\}.\]
    \item[(b)] Let the horizontal coordinate of $T^{(i)}_{j,k}$ be $h(i,j,k) \coloneqq in + j - (n+k)/2$. If $T^{(i)}_{j,k}=T^{(i')}_{j',k}=a$ for some $i,j,i',j',k$ with $h(i,j,k)<h(i',j',k)$, then there must exist $i'',j''$ with $T^{(i'')}_{j'',k-1}=a$ and $h(i,j,k)<h(i'',j'',k-1)<h(i',j',k)$. This entry $T^{(i'')}_{j'',k-1}$ is said to \emph{interlace} with $T^{(i)}_{j,k}$ and $T^{(i')}_{j',k}$.
\end{itemize}

For each $k\in[n]$, view $T^{(\bullet)}_{\bullet,k} \coloneqq \{T^{(i)}_{j,k} \mid 1 \leq i \leq m, 1 \leq j \leq k\}$ as the rows of an array of $m$ triangles, from bottom to top. Denote by $\T_{m,n}$ the set of interlacing triangular arrays of rank $m$ and height $n$ and by $\T_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$ the subset whose top row (that is, the row $k=n$) consists of $\lambda^{(1)},\ldots,\lambda^{(m)}$; here $\lambda^{(i)}\in[m]^n$ for $i\in[m]$. For $T\in\T_{m,n}$, use $T^{(i)}$ to denote the $i$-th triangle from left to right, and use $T^{(i)}_{\bullet,k}$ to denote its $k$-th row. 
\end{defin}
See \Cref{fig:interlacing-example} for an interlacing trangular array of rank $3$ and height $4$.

\begin{figure}[!ht]
    \centering
\begin{tikzpicture}[scale=1]
\def\sep{1.50000000000000}
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(3.00000000000000,0.000000000000000);
\draw[green!80,thin](3.00000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](0.000000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(2.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.00000000000000,0)--(2.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(2.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.00000000000000,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,0)--(2.50000000000000,-0.866000000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](0.000000000000000,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](3.00000000000000,0)--(3.00000000000000,0.000000000000000);
\draw[green!80,thin](4.50000000000000,0.000000000000000)--(7.50000000000000,0.000000000000000);
\draw[green!80,thin](7.50000000000000,0)--(6.00000000000000,-2.59800000000000);
\draw[green!80,thin](4.50000000000000,0)--(6.00000000000000,-2.59800000000000);
\draw[green!80,thin](5.00000000000000,-0.866000000000000)--(7.00000000000000,-0.866000000000000);
\draw[green!80,thin](6.50000000000000,0)--(5.50000000000000,-1.73200000000000);
\draw[green!80,thin](5.50000000000000,0)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](5.50000000000000,-1.73200000000000)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](5.50000000000000,0)--(5.00000000000000,-0.866000000000000);
\draw[green!80,thin](6.50000000000000,0)--(7.00000000000000,-0.866000000000000);
\draw[green!80,thin](6.00000000000000,-2.59800000000000)--(6.00000000000000,-2.59800000000000);
\draw[green!80,thin](4.50000000000000,0)--(4.50000000000000,0.000000000000000);
\draw[green!80,thin](7.50000000000000,0)--(7.50000000000000,0.000000000000000);
\draw[green!80,thin](9.00000000000000,0.000000000000000)--(12.0000000000000,0.000000000000000);
\draw[green!80,thin](12.0000000000000,0)--(10.5000000000000,-2.59800000000000);
\draw[green!80,thin](9.00000000000000,0)--(10.5000000000000,-2.59800000000000);
\draw[green!80,thin](9.50000000000000,-0.866000000000000)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](11.0000000000000,0)--(10.0000000000000,-1.73200000000000);
\draw[green!80,thin](10.0000000000000,0)--(11.0000000000000,-1.73200000000000);
\draw[green!80,thin](10.0000000000000,-1.73200000000000)--(11.0000000000000,-1.73200000000000);
\draw[green!80,thin](10.0000000000000,0)--(9.50000000000000,-0.866000000000000);
\draw[green!80,thin](11.0000000000000,0)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](10.5000000000000,-2.59800000000000)--(10.5000000000000,-2.59800000000000);
\draw[green!80,thin](9.00000000000000,0)--(9.00000000000000,0.000000000000000);
\draw[green!80,thin](12.0000000000000,0)--(12.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$1$};
\node at (1.00000000000000,-0.000000000000000) {$2$};
\node at (2.00000000000000,-0.000000000000000) {$1$};
\node at (3.00000000000000,-0.000000000000000) {$3$};
\node at (0.500000000000000,-0.866000000000000) {$1$};
\node at (1.50000000000000,-0.866000000000000) {$2$};
\node at (2.50000000000000,-0.866000000000000) {$1$};
\node at (1.00000000000000,-1.73200000000000) {$1$};
\node at (2.00000000000000,-1.73200000000000) {$2$};
\node at (1.50000000000000,-2.59800000000000) {$1$};
\node at (4.50000000000000,-0.000000000000000) {$1$};
\node at (5.50000000000000,-0.000000000000000) {$3$};
\node at (6.50000000000000,-0.000000000000000) {$3$};
\node at (7.50000000000000,-0.000000000000000) {$2$};
\node at (5.00000000000000,-0.866000000000000) {$3$};
\node at (6.00000000000000,-0.866000000000000) {$3$};
\node at (7.00000000000000,-0.866000000000000) {$1$};
\node at (5.50000000000000,-1.73200000000000) {$3$};
\node at (6.50000000000000,-1.73200000000000) {$1$};
\node at (6.00000000000000,-2.59800000000000) {$3$};
\node at (9.00000000000000,-0.000000000000000) {$1$};
\node at (10.0000000000000,-0.000000000000000) {$2$};
\node at (11.0000000000000,-0.000000000000000) {$3$};
\node at (12.0000000000000,-0.000000000000000) {$2$};
\node at (9.50000000000000,-0.866000000000000) {$2$};
\node at (10.5000000000000,-0.866000000000000) {$3$};
\node at (11.5000000000000,-0.866000000000000) {$2$};
\node at (10.0000000000000,-1.73200000000000) {$3$};
\node at (11.0000000000000,-1.73200000000000) {$2$};
\node at (10.5000000000000,-2.59800000000000) {$2$};
\end{tikzpicture}
    \caption{An interlacing triangular array $T$ of rank $3$ and height $4$. Throughout the paper, we draw interlacing triangular arrays in green ($\textcolor{green!80}{\blacksquare}$), $1/2/3$-puzzles in blue ($\textcolor{blue!80}{\blacksquare}$), $0/1/10$-puzzles in cyan ($\textcolor{cyan!80}{\blacksquare}$), and vertex-colored graphs in red ($\textcolor{red!80}{\blacksquare}$).
    }
    \label{fig:interlacing-example}
\end{figure}

%For integers $m,n \geq 1$ and tuples $\lambda^{(i)}\in [m]^n$ for $i=1,\ldots,m$, we write $\T_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$ for the set of interlacing triangular arrays of rank $m$ and height $n$ with top row $\lambda^{(1)},\ldots,\lambda^{(m)}$. We write $\T_{m,n}$ for $\bigsqcup_{\bs{\lambda}} \T_{m,n}(\bs{\lambda})$.
\subsection{Interlacing triangular arrays and graph colorings}
The following conjecture was our original motivation for this work.

\begin{conj}[Conjecture~A.3 of Aggarwal--Borodin--Wheeler \cite{Aggarwal-Borodin-Wheeler}]
\label{conj:3-color-conjecture}   
For $n \geq 1$, we have
\[
|\T_{3,n}|=\frac{1}{4} \left|\{\text{proper vertex $4$-colorings of } \Delta_n \} \right|,
\]
where $\Delta_n$ denotes the equilateral triangular grid graph with $n$ edges on each side.
\end{conj}

Our first main theorem resolves and significantly refines and extends \Cref{conj:3-color-conjecture}.

\begin{theorem}
\label{thm:intro-rank-3-bijections}
Fix $\lambda^{(1)},\lambda^{(2)},\lambda^{(3)} \in [3]^n$. Then the following sets of objects are in bijection:
\begin{itemize}
    \item[(1)] Interlacing triangular arrays $\T_{3,n}(\bs{\lambda})$ of rank $3$ with top row $\bs{\lambda}$;
    \item[(2)] $1/2/3$-puzzles $\mc{P}_n(\bs{\lambda})$ with boundary conditions $\bs{\lambda}$;
    \item[(3)] $0/1/10$-puzzles $\tilde{\mc{P}}_n(\bs{\xi})$ with boundary conditions $\bs{\xi}=\partn(\bs{\lambda})$;
    \item[(4)] Proper vertex $4$-colorings $\mc{C}_n(\bs{\kappa})$ of $\Delta_n$ with boundary colors $\bs{\kappa}=\colr({\bs{\lambda}})$.
\end{itemize}
\end{theorem}

The functions $\partn, \colr,$ and $\toprow$ are straightforward conversions between the different kinds of indexing data. The colorings $\bigsqcup_{\bs{\kappa}=\colr(\bs{\lambda})} \mc{C}_n(\bs{\kappa})$ from \Cref{thm:intro-rank-3-bijections}(4) are exactly those proper vertex $4$-colorings of $\Delta_n$ in which a fixed
base vertex is colored with the first color. The number of these is one fourth the total number of proper vertex $4$-colorings, so \Cref{thm:intro-rank-3-bijections} implies \Cref{conj:3-color-conjecture}. 

The particular bijections underlying \Cref{thm:intro-rank-3-bijections} allow us to give geometric interpretations for certain sets of interlacing triangular arrays (see \Cref{sec:intro-geometry}).

Aggarwal, Borodin, and Wheeler also conjectured a connection between interlacing triangular arrays of rank $4$ and graph colorings. Let $\boxtimes_n$ be the graph obtained from the $n \times n$ square grid graph $\square_n$ by adding the two diagonal edges of each face of $\square_n$.

\begin{conj}[Conjecture~A.5 of Aggarwal--Borodin--Wheeler \cite{Aggarwal-Borodin-Wheeler}]
\label{conj:4-color-conjecture-false}
For $n \geq 1$, we have
\begin{equation}
\label{eq:rank-4-coloring-conjecture-false}
    |\T_{4,n}| = \frac{1}{5} \left|\{\text{proper vertex $5$-colorings of } \boxtimes_n \} \right|.
\end{equation}
\end{conj}

In \Cref{thm:intro-rank-4-bijections} we give a bijection between interlacing triangular arrays of rank $4$ and certain edge labelings of the square grid graph $\square_n$, which are analogous to the $1/2/3$-puzzles of \Cref{thm:intro-rank-3-bijections}(2).

\begin{theorem}
\label{thm:intro-rank-4-bijections}
Fix $\lambda^{(1)},\lambda^{(2)},\lambda^{(3)},\lambda^{(4)} \in [4]^n$. Then the followings are in bijection:
\begin{itemize}
    \item[(1)] Interlacing triangular arrays $\T_{4,n}(\bs{\lambda})$ of rank $4$ with top row $\bs{\lambda}$;
    \item[(2)] Edge labelings $\D_n(\bs{\lambda})$ of the square grid graph $\square_n$ having boundary conditions $\bs{\lambda}$ and satisfying the conditions of \Cref{sec:squares}.
\end{itemize}
\end{theorem}

Unlike in the rank-$3$ case, there is no straightforward way to biject the objects in \Cref{thm:intro-rank-4-bijections} with proper \textit{vertex} colorings. In particular, by enumerating $\D_4$ we show that \Cref{conj:4-color-conjecture-false} is false\footnote{Leonid Petrov has independently observed the failure of \Cref{conj:4-color-conjecture-false} (personal communication).}, as
\[
|\T_{4,4}| = 191232 \neq 187008 = \frac{1}{5} \left|\{\text{proper vertex $5$-colorings of } \boxtimes_4 \} \right|.
\]


\subsection{Geometric interpretations of interlacing triangular arrays}
\label{sec:intro-geometry}

The $0/1/10$-puzzles appearing in \Cref{thm:intro-rank-3-bijections}(3) are known to have various geometric interpretations when certain puzzle pieces are forbidden and when the boundary conditions are appropriate. The number of such puzzles with boundary conditions $\bs{\xi}=(\xi^{(1)},\xi^{(2)},\xi^{(3)})$ computes the coefficient of the basis element indexed by $\xi^{(3)}$ in the product of basis elements indexed by $\xi^{(1)}$ and $\xi^{(2)}$ in cohomology $H^*(\gr(d,n))$ and $K$-theory $K(\gr(d,n))$ of Grassmannians, in the (appropriately localized) cohomology $H^{*\text{loc}}_{\C^{\times}}(T^*\gr(d,n))$ of their cotangent bundles, and for the multiplication in the cohomology of the 2-step flag variety \cite{Knutson-Tao-Woodward, Knutson-Zinn-Justin-2, Vakil,wheeler-zinn-justin}. 

We use the specific bijections underlying \Cref{thm:intro-rank-3-bijections} to show that interlacing triangular arrays with forbidden patterns and specified top row likewise compute these coefficients. One advantage of interlacing triangular arrays is that they allow for an interpretation of coefficients in the expansion of an $(m-1)$-fold product, without the need to iteratively apply a rule for products of two elements.

%We say that a top row $\bs{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)}) \in ([m]^n)^m$ is \emph{$(d,n)$-Grassmannian} if $\lambda^{(i)}$ has type $(m-i)^d(m-i+1)^{n-d}$ for $i=1,\ldots,m$.

For $\xi$ a $0,1$-string with content $0^d1^{n-d}$, let $G_{\xi}$ denote the class of the structure sheaf of the Schubert variety $X_{\xi} \subset \gr(d,n)$ inside $K(\gr(d,n))$. These classes can be represented by the \textit{(Grassmannian) Grothendieck polynomials}. The $\{G_{\xi}\}$ form a basis for $K(\gr(d,n))$. In \Cref{thm:k-theory} we show that the structure constants for multiplication in the basis $\{G_{\xi}\}$ are equal (up to signs) to the number of certain interlacing triangular arrays. Even the positivity of these structure constants (up to predictable signs) is not obvious, and is due originally to Buch \cite{Buch}.

Given $\bs{\xi}=(\xi^{(1)},\ldots,\xi^{(m)})$ of $0,1$-strings, define $|\bs{\xi}|=\sum_i |\xi^{(i)}|$, where $|\xi^{(i)}|$ is the number of inversions of $\xi^{(i)}$. For $\xi$ of content $0^d1^{n-d}$, denote by $\xi^{\perp}$ the reversed string.

\begin{theorem}
\label{thm:k-theory}
Let $\xi^{(1)},\ldots,\xi^{(m)}$ have content $0^d1^{n-d}$. Let coefficients $g_{\bs{\xi}}=g_{\xi^{(1)},\ldots,\xi^{(m)}}$ be determined by 
\[
\prod_{i=1}^{m-1} G_{\xi^{(i)}} = \sum_{\xi^{(m)}} g_{\bs{\xi}} G_{(\xi^{(m)})^{\perp}}.
\]
Then $(-1)^{d(n-d)-|\bs{\xi}|}g_{\bs{\xi}}$ is the number of interlacing triangular arrays $T$ from $\T_{m,n}(\toprow(\bs{\xi}))$ such that, for $i=2,\ldots,m-1$, $T^{(i)}$ avoids
\begin{equation}
\label{eq:k-theory-thm-forbidden}
\raisebox{-0.5\height}{
\begin{tikzpicture}
    \draw[green!80,thin] (0,0)--(-0.5,.866);
    \node at (-0.5,.866) {$m-i$};
    \node at (0,0) {$m$};
\end{tikzpicture}} \qquad and \qquad
\raisebox{-0.5\height}{
\begin{tikzpicture}
    \draw[green!80,thin] (0,0)--(-0.5,-.866)--(0.5,-.866)--(0,0);
    \node at (-0.75,-.866) {$m-i+1$};
    \node at (0.6,-.866) {$m$};
    \node at (0,0) {$m$};
\end{tikzpicture}.}
\end{equation}
\end{theorem}

Dual to $\{G_{\xi}\}$ is the basis $\{G^{\ast}_{\xi}\}$ of ideal sheaves: functions on Schubert varieties vanishing on smaller Schubert varieties. 
%The $G^{\ast}_{\xi}$ can be represented by \textit{dual Grothendieck polynomials}.


\begin{theorem}
\label{thm:dual-k}
Let $\xi^{(1)},\ldots,\xi^{(m)}$ have content $0^d1^{n-d}$. Let coefficients $g^{\ast}_{\bs{\xi}}=g^{\ast}_{\xi^{(1)},\ldots,\xi^{(m)}}$ be determined by 
\[
\prod_{i=1}^{m-1} G^{\ast}_{\xi^{(i)}} = \sum_{\xi^{(m)}} g^{\ast}_{\bs{\xi}} G^{\ast}_{(\xi^{(m)})^{\perp}}.
\]
Then $(-1)^{d(n-d)-|\bs{\xi}|}g^{\ast}_{\bs{\xi}}$ is the number of interlacing triangular arrays $T$ from $\T_{m,n}(\toprow(\bs{\xi}))$ such that, for $i=2,\ldots,m-1$, $T^{(i)}$ avoids 
\begin{equation}
\label{eq:dual-k-thm-forbidden}
\raisebox{-0.5\height}{
\begin{tikzpicture}
    \draw[green!80,thin] (0,0)--(-0.5,.866);
    \node at (-0.5,.866) {$m$};
    \node at (0,0) {$m-i$};
\end{tikzpicture}} \qquad and \qquad
\raisebox{-0.5\height}{
\begin{tikzpicture}
    \draw[green!80,thin] (0,0)--(-0.5,.866)--(0.5,.866)--(0,0);
    \node at (-0.6,.866) {$m$};
    \node at (0.75,.866) {$m-i+1$};
    \node at (0,0) {$m$};
\end{tikzpicture}.}
\end{equation}
\end{theorem}

Interlacing triangular arrays avoiding \emph{both} the patterns (\ref{eq:k-theory-thm-forbidden}) and (\ref{eq:dual-k-thm-forbidden}) compute structure constants in the ordinary cohomology of the Grassmannian. We can generalize this case to products of certain classes in the cohomology of arbitrary partial flag varieties.

For $\d = (0=d_0 \leq d_1 \leq \cdots \leq d_m=n)$, let $\Fl(\bs{d};n)$ denote the partial flag variety of flags of subspaces of $\C^n$ with dimension vector $\d$. Let $S_n^{\d}$ denote the set of permutations whose descents are contained in $\d$. Then $H^{\ast}(\Fl(\d;n))$ has a basis $\{\sigma_{w}\}_{w \in S_n^{\d}}$ consisting of the classes of the Schubert varieties in $\Fl(\d;n)$. In particular, the class $\sigma_{w_0^{\d}}$ of the longest element of $S_n^{\d}$ is the class of a point. For $w\in S_n^{\d}$, write $w^{\vee_{\d}} \coloneqq w_0 w w_0(\d)\in S_n^{\d}$. For $w^{(1)},\ldots,w^{(m)} \in S_n^{\d}$, let coefficients $c_{\bs{w}}=c_{w^{(1)},\ldots,w^{(m)}}$ be determined by
\[
\prod_{i=1}^{m-1} \sigma_{w^{(i)}} = \sum_{w^{(m)}} c_{\bs{w}} \sigma_{(w^{(m)})^{\vee_{\d}}},
\]


For a finite alphabet with a total order $\Sigma=\{q_1<\cdots <q_k\}$, we say that $\lambda$ has \emph{type} $q^{\alpha}=q_1^{\alpha_1}\cdots q_k^{\alpha_k}$ if $\lambda$ contains $\alpha_i$ copies of $q_i$, for $i=1,\ldots,k$. For such a string, we associate a permutation $w(\lambda)_{\Sigma}=w\in S_n$ such that $w(d_{i-1}+1)<\cdots<w(d_{i})$ are the positions of $q_i$'s in $\lambda$, where $\alpha=\alpha(\bs{d})$ is defined by $\alpha_1+\cdots+\alpha_{i}=d_i$ for $i=1,\ldots,k$.

\begin{theorem}
\label{thm:2-step}
Let $\d = (0=d_0 \leq d_1 \leq \cdots \leq d_m=n)$. For $i \in [m]$ let $\Sigma_i=\{m-i<m<m-i+1\}$, let $\lambda^{(i)}$ be a string of type $(m-i)^{d_{m-i}}m^{d_{m-i+1}-d_{m-i}}(m-i+1)^{n-d_{m-i+1}}$, and let $w^{(i)}=w(\lambda^{(i)})_{\Sigma_i}\in S_n^{\d}$. Then $c_{\bs{w}}$ is the number of interlacing triangular arrays $T$ from $\T_{m,n}(\bs{\lambda})$ such that, for $i=2,
\ldots,m-1$, $T^{(i)}$ avoids the patterns from (\ref{eq:k-theory-thm-forbidden}) and (\ref{eq:dual-k-thm-forbidden}).
\end{theorem}
The Schubert classes $\sigma_{w^{(i)}}$ appearing in \Cref{thm:2-step} are the pullbacks of Schubert classes under the projection from $\Fl(\d;n)$ to certain $2$-step flag varieties.


Finally, let $H^{*\text{loc}}_{\C^{\times}}(T^*\gr(d,n))$ denote the equivariant cohomology of the cotangent bundle of $\gr(d,n)$ with respect to the $\C^{\times}$-action scaling the cotangent spaces, localized as in \cite[Section~2.2]{Knutson-Zinn-Justin-2}. For $\xi$ of content $0^d1^{n-d}$, let $S_{\xi} \in H^{*\text{loc}}_{\C^{\times}}(T^*\gr(d,n))$ denote the \emph{Segre--Schwartz--MacPherson (SSM) class} of the corresponding Schubert variety, using the conventions of \cite[Sections~2.4 and 5.2]{Knutson-Zinn-Justin-2}. In our last main theorem, we show that interlacing triangular arrays, with \emph{no} forbidden patterns, compute structure constants for the $\{S_{\xi}\}$.

\begin{theorem}
\label{thm:ssm}
Let $\xi^{(1)},\ldots,\xi^{(m)}$ have content $0^d1^{n-d}$. Let coefficients $s_{\bs{\xi}}=s_{\xi^{(1)},\ldots,\xi^{(m)}}$ be determined by 
\[
\prod_{i=1}^{m-1} S_{\xi^{(i)}} = \sum_{\xi^{(m)}} s_{\bs{\xi}} S_{(\xi^{(m)})^{\perp}}.
\]
Then $(-1)^{d(n-d)-|\bs{\xi}|}s_{\bs{\xi}}$ is the cardinality of $\T_{m,n}(\toprow(\bs{\xi}))$.
\end{theorem}

\subsection{Examples of the geometric interpretations}

\begin{ex}\label{ex:grassmannian-}
Let $m=4$ and $n=4$. Consider the $0,1$-strings $\xi^{(1)}=\xi^{(2)}=\xi^{(3)}=0101$ with length $1$. Correspondingly, $\lambda^{(1)}=\toprow(\xi^{(1)})=3434$, and analogously, $\lambda^{(2)}=2323$ and $\lambda^{(3)}=1212$. They also correspond to the permutation $1324$ and the partition with one box. As an example of \Cref{thm:2-step}, in $H^*(\gr(2,4))$, $\sigma_{1324}^3=2\sigma_{2413}$, whose $0,1$-string is $1010$ and the coefficient $2$ counts the interlacing triangular arrays in \Cref{fig:Gr24-interlacing-example}.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$3$};
\node at (7.5,-0.866) {$2$};
\node at (6.0,-1.732) {$3$};
\node at (7.0,-1.732) {$2$};
\node at (6.5,-2.598) {$4$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$1$};
\node at (11.5,-0.866) {$4$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$1$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$1$};
\node at (17.0,-0.0) {$4$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$1$};
\node at (16.5,-0.866) {$4$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$4$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}

\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$4$};
\node at (7.5,-0.866) {$3$};
\node at (6.0,-1.732) {$2$};
\node at (7.0,-1.732) {$3$};
\node at (6.5,-2.598) {$2$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$2$};
\node at (12.0,-1.732) {$1$};
\node at (11.5,-2.598) {$4$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$1$};
\node at (17.0,-0.0) {$4$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$1$};
\node at (16.5,-0.866) {$4$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$4$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\caption{Interlacing triangular arrays with top row $\lambda^{(1)}=3434,\lambda^{(2)}=2323,\lambda^{(3)}=1212,\lambda^{(4)}=4141$.}
\label{fig:Gr24-interlacing-example}
\end{figure}

There are more interlacing triangular arrays whose top row starts with $\lambda^{(1)} = 3434$, $\lambda^{(2)} = 2323$, $\lambda^{(3)} = 1212$. Besides the ones in \Cref{fig:Gr24-interlacing-example}, all the others have top row $\lambda^{(4)}=4411$ shown in \Cref{fig:Gr24-interlacing-example-more}. The first five of them contain patterns from (\ref{eq:k-theory-thm-forbidden}) and the last one contains patterns from (\ref{eq:dual-k-thm-forbidden}), highlighted in the figure. 
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$3$};
\node at (7.5,-0.866) {$2$};
\node at (6.0,-1.732) {$3$};
\node at (7.0,-1.732) {$2$};
\node at (6.5,-2.598) {$4$};
\draw (10.0,-0.0) circle [radius=0.4];
\draw (10.5,-0.866) circle [radius=0.4];
\draw[ultra thick,green](10.0,-0.0)--(10.5,-0.866);
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$4$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$4$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\draw[ultra thick,green] (7.5,-0.866)--(7.0,-0.0);
\draw (7.5,-0.866) circle [radius=0.4];
\draw (7.0,-0.0) circle [radius=0.4];
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$3$};
\node at (7.5,-0.866) {$4$};
\node at (6.0,-1.732) {$3$};
\node at (7.0,-1.732) {$2$};
\node at (6.5,-2.598) {$4$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$4$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$4$};
\node at (7.5,-0.866) {$3$};
\node at (6.0,-1.732) {$2$};
\node at (7.0,-1.732) {$3$};
\node at (6.5,-2.598) {$2$};
\draw[ultra thick,green] (11.5,-0.866)--(12.0,-1.732);
\draw (11.5,-0.866) circle [radius=0.4];
\draw (12.0,-1.732) circle [radius=0.4];
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$2$};
\node at (12.0,-1.732) {$4$};
\node at (11.5,-2.598) {$4$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\draw[ultra thick,green] (5.0,-0.0)--(5.5,-0.866);
\draw (5.0,-0.0) circle [radius=0.4];
\draw (5.5,-0.866) circle [radius=0.4];
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$4$};
\node at (6.5,-0.866) {$2$};
\node at (7.5,-0.866) {$3$};
\node at (6.0,-1.732) {$3$};
\node at (7.0,-1.732) {$2$};
\node at (6.5,-2.598) {$4$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$4$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\draw[ultra thick,green] (6.0,-1.732)--(6.5,-2.598);
\draw (6.0,-1.732) circle [radius=0.4];
\draw (6.5,-2.598) circle [radius=0.4];
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$4$};
\node at (7.5,-0.866) {$3$};
\node at (6.0,-1.732) {$2$};
\node at (7.0,-1.732) {$3$};
\node at (6.5,-2.598) {$4$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$4$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\def\sep{2}
\draw[green,thin](0.0,0.0)--(3.0,0.0);
\draw[green,thin](3,0)--(1.5,-2.598);
\draw[green,thin](0,0)--(1.5,-2.598);
\draw[green,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green,thin](2,0)--(1.0,-1.7319999999999998);
\draw[green,thin](1,0)--(2.0,-1.7319999999999998);
\draw[green,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green,thin](1,0)--(0.5,-0.8659999999999999);
\draw[green,thin](2,0)--(2.5,-0.8659999999999999);
\draw[green,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green,thin](0,0)--(0.0,0.0);
\draw[green,thin](3,0)--(3.0,0.0);
\draw[green,thin](5.0,0.0)--(8.0,0.0);
\draw[green,thin](8,0)--(6.5,-2.598);
\draw[green,thin](5,0)--(6.5,-2.598);
\draw[green,thin](5.5,-0.866)--(7.5,-0.866);
\draw[green,thin](7,0)--(6.0,-1.7319999999999998);
\draw[green,thin](6,0)--(7.0,-1.7319999999999998);
\draw[green,thin](6.0,-1.732)--(7.0,-1.732);
\draw[green,thin](6,0)--(5.5,-0.8659999999999999);
\draw[green,thin](7,0)--(7.5,-0.8659999999999999);
\draw[green,thin](6.5,-2.598)--(6.5,-2.598);
\draw[green,thin](5,0)--(5.0,0.0);
\draw[green,thin](8,0)--(8.0,0.0);
\draw[green,thin](10.0,0.0)--(13.0,0.0);
\draw[green,thin](13,0)--(11.5,-2.598);
\draw[green,thin](10,0)--(11.5,-2.598);
\draw[green,thin](10.5,-0.866)--(12.5,-0.866);
\draw[green,thin](12,0)--(11.0,-1.7319999999999998);
\draw[green,thin](11,0)--(12.0,-1.7319999999999998);
\draw[green,thin](11.0,-1.732)--(12.0,-1.732);
\draw[green,thin](11,0)--(10.5,-0.8659999999999999);
\draw[green,thin](12,0)--(12.5,-0.8659999999999999);
\draw[green,thin](11.5,-2.598)--(11.5,-2.598);
\draw[green,thin](10,0)--(10.0,0.0);
\draw[green,thin](13,0)--(13.0,0.0);
\draw[green,thin](15.0,0.0)--(18.0,0.0);
\draw[green,thin](18,0)--(16.5,-2.598);
\draw[green,thin](15,0)--(16.5,-2.598);
\draw[green,thin](15.5,-0.866)--(17.5,-0.866);
\draw[green,thin](17,0)--(16.0,-1.7319999999999998);
\draw[green,thin](16,0)--(17.0,-1.7319999999999998);
\draw[green,thin](16.0,-1.732)--(17.0,-1.732);
\draw[green,thin](16,0)--(15.5,-0.8659999999999999);
\draw[green,thin](17,0)--(17.5,-0.8659999999999999);
\draw[green,thin](16.5,-2.598)--(16.5,-2.598);
\draw[green,thin](15,0)--(15.0,0.0);
\draw[green,thin](18,0)--(18.0,0.0);
\node at (0.0,-0.0) {$3$};
\node at (1.0,-0.0) {$4$};
\node at (2.0,-0.0) {$3$};
\node at (3.0,-0.0) {$4$};
\node at (0.5,-0.866) {$3$};
\node at (1.5,-0.866) {$4$};
\node at (2.5,-0.866) {$3$};
\node at (1.0,-1.732) {$3$};
\node at (2.0,-1.732) {$4$};
\node at (1.5,-2.598) {$3$};
\draw[ultra thick,green] (6.5,-0.866)--(7.0,-1.732);
\draw (6.5,-0.866) circle [radius=0.4];
\draw (7.0,-1.732) circle [radius=0.4];
\node at (5.0,-0.0) {$2$};
\node at (6.0,-0.0) {$3$};
\node at (7.0,-0.0) {$2$};
\node at (8.0,-0.0) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$4$};
\node at (7.5,-0.866) {$3$};
\node at (6.0,-1.732) {$3$};
\node at (7.0,-1.732) {$2$};
\node at (6.5,-2.598) {$4$};
\node at (10.0,-0.0) {$1$};
\node at (11.0,-0.0) {$2$};
\node at (12.0,-0.0) {$1$};
\node at (13.0,-0.0) {$2$};
\node at (10.5,-0.866) {$2$};
\node at (11.5,-0.866) {$1$};
\node at (12.5,-0.866) {$2$};
\node at (11.0,-1.732) {$4$};
\node at (12.0,-1.732) {$2$};
\node at (11.5,-2.598) {$2$};
\node at (15.0,-0.0) {$4$};
\node at (16.0,-0.0) {$4$};
\node at (17.0,-0.0) {$1$};
\node at (18.0,-0.0) {$1$};
\node at (15.5,-0.866) {$4$};
\node at (16.5,-0.866) {$1$};
\node at (17.5,-0.866) {$1$};
\node at (16.0,-1.732) {$1$};
\node at (17.0,-1.732) {$1$};
\node at (16.5,-2.598) {$1$};
\end{tikzpicture}
\caption{Interlacing triangular arrays with top row $\lambda^{(1)}=3434,\lambda^{(2)}=2323,\lambda^{(3)}=1212,\lambda^{(4)}=4411$.}
\label{fig:Gr24-interlacing-example-more}
\end{figure}

Now, \Cref{thm:k-theory}, \Cref{thm:dual-k} and \Cref{thm:ssm} imply that
\[G_{0101}^3=2G_{1010}-G_{1100},\quad (G_{0101}^*)^3=2G_{1010}^*-5G_{1100}^*,\quad S_{0101}^3=2S_{1010}-6S_{1100}.\]
\end{ex}

\subsection{Outline}
In \Cref{sec:puzzle}, we prove \Cref{thm:psi-is-bijection}, establishing bijections between rank-$3$ interlacing triangular arrays $\T_{3,n}$ and $1/2/3$-puzzles $\mc{P}_n$. In \Cref{sec:puzzle-to-colorings} we in turn prove \Cref{thm:counting-m3}, giving bijections between $1/2/3$-puzzles $\mc{P}_n$ and proper vertex colorings $\mc{C}_n$ of $\Delta_n$. In \Cref{thm:counting-m4} we also give bijections between rank-$4$ interlacing triangular arrays $\T_{4,n}$ and certain edge labelings $\mc{D}_n$ of $\square_n$. Finally, in \Cref{sec:geometric-sketch} we sketch some key ideas that go into the proofs of the geometric interpretations in \Cref{thm:k-theory,thm:dual-k,thm:2-step,thm:ssm}. These include a correspondence between $1/2/3$-puzzles and $0/1/10$-puzzles for which we are grateful to Allen Knutson and a \emph{splitting lemma} on certain interlacing triangular arrays allowing us to reduce to the case $m=3$. 

%, for which we can utilize to trace through forbidden substructures in interlacing triangular arrays from forbidden one or both of the $\Delta$- and $\nabla$-oriented $10$-$10$-$10$ pieces in the geometric interpretations with respect to the $G_{\xi}$ basis in $K$-theory and the $G^{\ast}_{\xi}$ basis in $K$-theory.

% In \Cref{sec:geometry}, we show how to convert between $1/2/3$-puzzles and $0/1/10$-puzzles. Tracing this correspondence through to interlacing triangular arrays using the bijection $\mathscr{T}$, we show that forbidding certain of the $0/1/10$-puzzle pieces corresponds to forbidding certain patterns in the arrays. In \Cref{sec:splitting}, we prove the key \Cref{lem:splitting-interlacing} which allows us to split arrays into pairs of arrays of lower rank. Finally, these results are applied in \Cref{sec:geometry} to prove \Cref{thm:k-theory,thm:dual-k,thm:ssm,thm:2-step}.

\section{From interlacing triangular arrays to puzzles}
\label{sec:puzzle}

The goal of this section is to establish a bijection $\mathscr{T}$ between interlacing triangular arrays of rank $3$ and certain edge labelings of $\Delta_n$. We call these labelings \textit{$1/2/3$-puzzles} since they in turn are in bijection with the $0/1/10$-puzzles of Knutson--Tao \cite{Knutson-Tao-equivariant} as generalized by Knutson--Zinn-Justin \cite[Section 4]{Knutson-Zinn-Justin-2}.

\begin{defin}
\label{def:123-puzzles}
We denote by $\Delta_n$ the \emph{triangular grid graph} with side length $n$; see \Cref{fig:123-puzzle-example}. We view $\Delta_n$ as embedded in the plane as pictured, allowing us to distinguish between the $\Delta$-oriented and $\nabla$-oriented faces. We take the lower left corner as a distinguished base point, and view $\Delta_{n-1}$ as a subgraph of $\Delta_{n}$, sharing the base point.  

A \emph{$1/2/3$-puzzle} is a labeling of the edges of $\Delta_n$ with labels $1$, $2$, and $3$ so that each face has distinct edge labels. We write $\mc{P}_n$ for the set of these puzzles.

The \emph{boundary conditions} $\boldsymbol{\mu}=(\mu^{(1)},\mu^{(2)}, \mu^{(3)})$ of a puzzle $P \in \mc{P}_n$ are the labelings of the three sides of the triangle $\Delta_n$, read clockwise starting from the base point. We write $\mc{P}_n(\boldsymbol{\mu})$ for the set of puzzles from $\mc{P}_n$ with boundary conditions $\boldsymbol{\mu}$.
\end{defin}

\begin{figure}
\centering
    \begin{tikzpicture}[scale=0.95]
\draw[thin,blue!80](0,0)--(4,0)--(2.00000000000000,3.46400000000000)--(0,0);
\draw[thin,blue!80](0.500000000000000,0.866000000000000)--(3.50000000000000,0.866000000000000);
\draw[thin,blue!80](1,0)--(2.50000000000000,2.59800000000000);
\draw[thin,blue!80](0.500000000000000,0.866000000000000)--(1,0);
\draw[thin,blue!80](1.00000000000000,1.73200000000000)--(3.00000000000000,1.73200000000000);
\draw[thin,blue!80](2,0)--(3.00000000000000,1.73200000000000);
\draw[thin,blue!80](1.00000000000000,1.73200000000000)--(2,0);
\draw[thin,blue!80](1.50000000000000,2.59800000000000)--(2.50000000000000,2.59800000000000);
\draw[thin,blue!80](3,0)--(3.50000000000000,0.866000000000000);
\draw[thin,blue!80](1.50000000000000,2.59800000000000)--(3,0);



\node at (0.250000000000000,0.433000000000000) {$1$};
\node at (2.25000000000000,3.03100000000000) {$1$};
\node at (0.500000000000000,0.000000000000000) {$2$};
\node at (0.750000000000000,1.29900000000000) {$2$};
\node at (2.75000000000000,2.16500000000000) {$3$};
\node at (1.50000000000000,0.000000000000000) {$3$};
\node at (1.25000000000000,2.16500000000000) {$1$};
\node at (3.25000000000000,1.29900000000000) {$3$};
\node at (2.50000000000000,0.000000000000000) {$2$};
\node at (1.75000000000000,3.03100000000000) {$3$};
\node at (3.75000000000000,0.433000000000000) {$2$};
\node at (3.50000000000000,0.000000000000000) {$1$};
\node at (1.25000000000000,0.433000000000000) {$2$};
\node at (1.75000000000000,2.16500000000000) {$3$};
\node at (1.00000000000000,0.866000000000000) {$1$};
\node at (1.75000000000000,1.29900000000000) {$1$};
\node at (2.25000000000000,1.29900000000000) {$3$};
\node at (2.00000000000000,0.866000000000000) {$2$};
\node at (2.25000000000000,2.16500000000000) {$1$};
\node at (2.75000000000000,0.433000000000000) {$1$};
\node at (3.00000000000000,0.866000000000000) {$2$};
\node at (2.25000000000000,0.433000000000000) {$3$};
\node at (1.25000000000000,1.29900000000000) {$3$};
\node at (1.50000000000000,1.73200000000000) {$2$};
\node at (2.75000000000000,1.29900000000000) {$1$};
\node at (1.75000000000000,0.433000000000000) {$1$};
\node at (2.50000000000000,1.73200000000000) {$2$};
\node at (3.25000000000000,0.433000000000000) {$3$};
\node at (0.750000000000000,0.433000000000000) {$3$};
\node at (2.00000000000000,2.59800000000000) {$2$};
\end{tikzpicture}
\caption{A $1/2/3$-puzzle $P$ with boundary conditions $(1213, 1332, 1232)$.}
\label{fig:123-puzzle-example}
\end{figure}

\begin{defin}
\label{def:puzzle-to-array-map}
Given a $1/2/3$-puzzle $P \in \mc{P}_n$, we produce a collection 
\[
\mathscr{T}(P) = \{T^{(i)}_{j,k} \mid 1 \leq i \leq 3, 1 \leq j \leq k \leq n \}
\]
of integers from $\{1,2,3\}$ as follows. For each $k=1,\ldots,n$, consider the copy of $\Delta_k$ inside $\Delta_n$ justified into the lower left corner of $\Delta_n$. The $k$-th row of $\mathscr{T}(P)$ is obtained by reading the labels of $P$ clockwise around $\Delta_k$, starting from the lower left vertex.
\end{defin}

\begin{ex}
The $1/2/3$-puzzle $P$ from \Cref{fig:123-puzzle-example} is sent by $\mathscr{T}$ to the array $T$ from \Cref{fig:interlacing-example}.
\end{ex}

\begin{theorem}
\label{thm:psi-is-bijection}
For any $P \in \mc{P}_n$, the array $\mathscr{T}(P)$ is an interlacing triangular array of rank $3$ and height $n$. Furthermore, map $\mathscr{T}$ is a bijection $\mc{P}_n \to \mc{T}_{3,n}$ restricting, for each boundary condition $\boldsymbol{\lambda}$, to a bijection $\mc{P}_n(\boldsymbol{\lambda} )\to \mc{T}_{3,n}(\boldsymbol{\lambda})$.
\end{theorem}


\section{From puzzles to graph colorings}
\label{sec:puzzle-to-colorings}
In this section, we relate interlacing triangular arrays and $123$-puzzles to graph colorings. We prove Conjecture~A.3 of \cite{Aggarwal-Borodin-Wheeler} and disprove Conjecture~A.5. We also formulate a new conjecture (\Cref{conj:counting-m4-proper-labeling}) on the enumeration of $\T_{4,n}$.

\subsection{\texorpdfstring{$\T_{3,n}$}{T(3,n)} and the triangular grid}
\label{sec:triangular-coloring}
Recall from \Cref{sec:puzzle} that $\Delta_n$ denotes the triangular grid graph with side length $n$, a graph on ${\binom{n+2}{2}}$ vertices. Let $\mc{C}_n$ denote the set of proper colorings of the vertices of $\Delta_n$ by $\{0,1,2,3\}$ such that the base point is colored $0$. We write $\mc{C}_n(\bs{\kappa})$ for the set of colorings from $\mc{C}_n$ with fixed coloring $\bs{\kappa}$ of the boundary vertices. We have proven in \Cref{thm:psi-is-bijection} that the map $\mathscr{T}$ is a bijection $\mathscr{T}:\mc{P}_n \to \mc{T}_{3,n}$. We now define a map $\mathscr{P}: \mc{C}_n \to \mc{P}_n$, proven in \Cref{thm:counting-m3} to be a bijection.

\begin{defin}
    Given a proper vertex coloring $C\in\mc{C}_n$, define an edge labeling $\mathscr{P}(C)$ as follows: for an edge $e$ of $\Delta_n$ incident to vertices $v$ and $v'$, set $\mathscr{P}(C)(e)=i$ if $\{c(v),c(v')\}=\{0,i\}$ or $\{0,1,2,3\}\setminus\{0,i\}$. See \Cref{fig:phi-example} for an example. By \Cref{thm:counting-m3} below, $\mathscr{P}$ is invertible on $1/2/3$-puzzles and so determines boundary colors $\colr(\bs{\lambda})$ given boundary conditions $\bs{\lambda}$ of a puzzle.
\end{defin}

\begin{theorem}
\label{thm:counting-m3}
The map $\mathscr{P}$ is a bijection $\mc{C}_n\rightarrow \mc{P}_n$ sending $\mc{C}(\colr(\bs{\lambda}))$ to $\mc{P}_n(\bs{\lambda})$.
\end{theorem}

\begin{figure}[!ht]
\centering
\raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.95]
\draw[thin,red!80](0,0)--(4,0)--(2.00000000000000,3.46400000000000)--(0,0);
\draw[thin,red!80](0.500000000000000,0.866000000000000)--(3.50000000000000,0.866000000000000);
\draw[thin,red!80](1,0)--(2.50000000000000,2.59800000000000);
\draw[thin,red!80](0.500000000000000,0.866000000000000)--(1,0);
\draw[thin,red!80](1.00000000000000,1.73200000000000)--(3.00000000000000,1.73200000000000);
\draw[thin,red!80](2,0)--(3.00000000000000,1.73200000000000);
\draw[thin,red!80](1.00000000000000,1.73200000000000)--(2,0);
\draw[thin,red!80](1.50000000000000,2.59800000000000)--(2.50000000000000,2.59800000000000);
\draw[thin,red!80](3,0)--(3.50000000000000,0.866000000000000);
\draw[thin,red!80](1.50000000000000,2.59800000000000)--(3,0);
\node at (0.000000000000000,0.000000000000000) {$0$};
\node at (1.00000000000000,0.000000000000000) {$2$};
\node at (2.00000000000000,0.000000000000000) {$1$};
\node at (3.00000000000000,0.000000000000000) {$3$};
\node at (4.00000000000000,0.000000000000000) {$2$};
\node at (0.500000000000000,0.866000000000000) {$1$};
\node at (1.50000000000000,0.866000000000000) {$0$};
\node at (2.50000000000000,0.866000000000000) {$2$};
\node at (3.50000000000000,0.866000000000000) {$0$};
\node at (1.00000000000000,1.73200000000000) {$3$};
\node at (2.00000000000000,1.73200000000000) {$1$};
\node at (3.00000000000000,1.73200000000000) {$3$};
\node at (1.50000000000000,2.59800000000000) {$2$};
\node at (2.50000000000000,2.59800000000000) {$0$};
\node at (2.00000000000000,3.46400000000000) {$1$};
\end{tikzpicture}}
\qquad $\xmapsto{\mathscr{P}}$ \qquad
\raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.95]
\draw[thin,blue!80](0,0)--(4,0)--(2.00000000000000,3.46400000000000)--(0,0);
\draw[thin,blue!80](0.500000000000000,0.866000000000000)--(3.50000000000000,0.866000000000000);
\draw[thin,blue!80](1,0)--(2.50000000000000,2.59800000000000);
\draw[thin,blue!80](0.500000000000000,0.866000000000000)--(1,0);
\draw[thin,blue!80](1.00000000000000,1.73200000000000)--(3.00000000000000,1.73200000000000);
\draw[thin,blue!80](2,0)--(3.00000000000000,1.73200000000000);
\draw[thin,blue!80](1.00000000000000,1.73200000000000)--(2,0);
\draw[thin,blue!80](1.50000000000000,2.59800000000000)--(2.50000000000000,2.59800000000000);
\draw[thin,blue!80](3,0)--(3.50000000000000,0.866000000000000);
\draw[thin,blue!80](1.50000000000000,2.59800000000000)--(3,0);



\node at (0.250000000000000,0.433000000000000) {$1$};
\node at (2.25000000000000,3.03100000000000) {$1$};
\node at (0.500000000000000,0.000000000000000) {$2$};
\node at (0.750000000000000,1.29900000000000) {$2$};
\node at (2.75000000000000,2.16500000000000) {$3$};
\node at (1.50000000000000,0.000000000000000) {$3$};
\node at (1.25000000000000,2.16500000000000) {$1$};
\node at (3.25000000000000,1.29900000000000) {$3$};
\node at (2.50000000000000,0.000000000000000) {$2$};
\node at (1.75000000000000,3.03100000000000) {$3$};
\node at (3.75000000000000,0.433000000000000) {$2$};
\node at (3.50000000000000,0.000000000000000) {$1$};
\node at (1.25000000000000,0.433000000000000) {$2$};
\node at (1.75000000000000,2.16500000000000) {$3$};
\node at (1.00000000000000,0.866000000000000) {$1$};
\node at (1.75000000000000,1.29900000000000) {$1$};
\node at (2.25000000000000,1.29900000000000) {$3$};
\node at (2.00000000000000,0.866000000000000) {$2$};
\node at (2.25000000000000,2.16500000000000) {$1$};
\node at (2.75000000000000,0.433000000000000) {$1$};
\node at (3.00000000000000,0.866000000000000) {$2$};
\node at (2.25000000000000,0.433000000000000) {$3$};
\node at (1.25000000000000,1.29900000000000) {$3$};
\node at (1.50000000000000,1.73200000000000) {$2$};
\node at (2.75000000000000,1.29900000000000) {$1$};
\node at (1.75000000000000,0.433000000000000) {$1$};
\node at (2.50000000000000,1.73200000000000) {$2$};
\node at (3.25000000000000,0.433000000000000) {$3$};
\node at (0.750000000000000,0.433000000000000) {$3$};
\node at (2.00000000000000,2.59800000000000) {$2$};
\end{tikzpicture}}
\caption{The bijection $\mathscr{P}$ from \Cref{thm:counting-m3}.}
\label{fig:phi-example}
\end{figure}

\subsection{\texorpdfstring{$\T_{4,n}$}{T(4,n)} and the square grid}
\label{sec:squares}

Let $\square_n$ be the square grid graph with side length $n$. For $k<n$, we view $\square_k$ as a southwest-justified subgraph of $\square_n$, and we fix the southwest corner as the basepoint. Let $\mc{D}_n$ be the set of edge labelings of $\square_n$ with four labels satisfying the following:
\begin{itemize}
\item for each face on the main SW-NE diagonal, all four edges are labeled differently;
\item for each of the off-diagonal faces, the four edges are labeled with exactly two distinct labels so that either one label is assigned to the west and south boundaries (with the other label assigned to the north and east boundaries), or one label is assigned to the north and south boundaries (with the other label assigned to the west and east boundaries).
\end{itemize}

We now define a map $\mathscr{D}'$ from $\mathcal{D}_n$ to certain triangular arrays of integers.

\begin{defin}
\label{def:D-to-T4}
Given an edge labeling $D\in\mc{D}_n$, let $\mathscr{D}'(D)$ be the triangular array of integers whose $k$-th row is obtained by reading the $D$-labels around the boundary of $\square_k$ in the clockwise direction, starting from the base point. See \Cref{fig:phi-example-m4} for an example.
\end{defin}

\begin{figure}[!ht]
\centering
\raisebox{-.5\height}{
\begin{tikzpicture}[scale=0.8]
\draw[thin,blue!80] (0,0)--(4,0);
\draw[thin,blue!80] (0,1)--(4,1);
\draw[thin,blue!80] (0,2)--(4,2);
\draw[thin,blue!80] (0,3)--(4,3);
\draw[thin,blue!80] (0,4)--(4,4);
\draw[thin,blue!80] (0,0)--(0,4);
\draw[thin,blue!80] (1,0)--(1,4);
\draw[thin,blue!80] (2,0)--(2,4);
\draw[thin,blue!80] (3,0)--(3,4);
\draw[thin,blue!80] (4,0)--(4,4);
\filldraw[blue!80,opacity=0.2] (0,0)--(0,1)--(1,1)--(1,0);
\filldraw[blue!80,opacity=0.2] (1,1)--(1,2)--(2,2)--(2,1);
\filldraw[blue!80,opacity=0.2] (2,2)--(2,3)--(3,3)--(3,2);
\filldraw[blue!80,opacity=0.2] (3,3)--(3,4)--(4,4)--(4,3);
\draw[thin,blue!80](0.500000000000000,1)--(0.500000000000000,2);
\draw[thin,blue!80](0,1.50000000000000)--(1,1.50000000000000);
\draw[thin,blue!80](0.500000000000000,2)--(0.500000000000000,3);
\draw[thin,blue!80](0,2.50000000000000)--(1,2.50000000000000);
\draw[thin,blue!80](0.500000000000000,3.20000000000000)arc(0:90:0.300000000000000);
\draw[thin,blue!80](0.500000000000000,3.80000000000000)arc(180:270:0.300000000000000);
\draw[thin,blue!80](0.500000000000000,3)--(0.500000000000000,3.20000000000000);
\draw[thin,blue!80](0.200000000000000,3.50000000000000)--(0,3.50000000000000);
\draw[thin,blue!80](0.500000000000000,4)--(0.500000000000000,3.80000000000000);
\draw[thin,blue!80](0.800000000000000,3.50000000000000)--(1,3.50000000000000);
\draw[thin,blue!80](1.50000000000000,0)--(1.50000000000000,1);
\draw[thin,blue!80](1,0.500000000000000)--(2,0.500000000000000);
\draw[thin,blue!80](1.50000000000000,2)--(1.50000000000000,3);
\draw[thin,blue!80](1,2.50000000000000)--(2,2.50000000000000);
\draw[thin,blue!80](1.50000000000000,3.20000000000000)arc(0:90:0.300000000000000);
\draw[thin,blue!80](1.50000000000000,3.80000000000000)arc(180:270:0.300000000000000);
\draw[thin,blue!80](1.50000000000000,3)--(1.50000000000000,3.20000000000000);
\draw[thin,blue!80](1.20000000000000,3.50000000000000)--(1,3.50000000000000);
\draw[thin,blue!80](1.50000000000000,4)--(1.50000000000000,3.80000000000000);
\draw[thin,blue!80](1.80000000000000,3.50000000000000)--(2,3.50000000000000);
\draw[thin,blue!80](2.50000000000000,0)--(2.50000000000000,1);
\draw[thin,blue!80](2,0.500000000000000)--(3,0.500000000000000);
\draw[thin,blue!80](2.50000000000000,1)--(2.50000000000000,2);
\draw[thin,blue!80](2,1.50000000000000)--(3,1.50000000000000);
\draw[thin,blue!80](2.50000000000000,3.20000000000000)arc(0:90:0.300000000000000);
\draw[thin,blue!80](2.50000000000000,3.80000000000000)arc(180:270:0.300000000000000);
\draw[thin,blue!80](2.50000000000000,3)--(2.50000000000000,3.20000000000000);
\draw[thin,blue!80](2.20000000000000,3.50000000000000)--(2,3.50000000000000);
\draw[thin,blue!80](2.50000000000000,4)--(2.50000000000000,3.80000000000000);
\draw[thin,blue!80](2.80000000000000,3.50000000000000)--(3,3.50000000000000);
\draw[thin,blue!80](3.50000000000000,0.200000000000000)arc(0:90:0.300000000000000);
\draw[thin,blue!80](3.50000000000000,0.800000000000000)arc(180:270:0.300000000000000);
\draw[thin,blue!80](3.50000000000000,0)--(3.50000000000000,0.200000000000000);
\draw[thin,blue!80](3.20000000000000,0.500000000000000)--(3,0.500000000000000);
\draw[thin,blue!80](3.50000000000000,1)--(3.50000000000000,0.800000000000000);
\draw[thin,blue!80](3.80000000000000,0.500000000000000)--(4,0.500000000000000);
\draw[thin,blue!80](3.50000000000000,1)--(3.50000000000000,2);
\draw[thin,blue!80](3,1.50000000000000)--(4,1.50000000000000);
\draw[thin,blue!80](3.50000000000000,2)--(3.50000000000000,3);
\draw[thin,blue!80](3,2.50000000000000)--(4,2.50000000000000);
\node at (0.5,0) {$1$};
\node at (1.5,0) {$3$};
\node at (2.5,0) {$1$};
\node at (3.5,0) {$4$};
\node at (0.5,1) {$3$};
\node at (1.5,1) {$3$};
\node at (2.5,1) {$1$};
\node at (3.5,1) {$1$};
\node at (0.5,2) {$3$};
\node at (1.5,2) {$2$};
\node at (2.5,2) {$1$};
\node at (3.5,2) {$1$};
\node at (0.5,3) {$3$};
\node at (1.5,3) {$2$};
\node at (2.5,3) {$3$};
\node at (3.5,3) {$1$};
\node at (0.5,4) {$2$};
\node at (1.5,4) {$3$};
\node at (2.5,4) {$4$};
\node at (3.5,4) {$2$};
\node at (0,0.5) {$2$};
\node at (0,1.5) {$1$};
\node at (0,2.5) {$4$};
\node at (0,3.5) {$3$};
\node at (1,0.5) {$4$};
\node at (1,1.5) {$1$};
\node at (1,2.5) {$4$};
\node at (1,3.5) {$2$};
\node at (2,0.5) {$4$};
\node at (2,1.5) {$4$};
\node at (2,2.5) {$4$};
\node at (2,3.5) {$3$};
\node at (3,0.5) {$4$};
\node at (3,1.5) {$4$};
\node at (3,2.5) {$2$};
\node at (3,3.5) {$4$};
\node at (4,0.5) {$1$};
\node at (4,1.5) {$4$};
\node at (4,2.5) {$2$};
\node at (4,3.5) {$3$};
\node at (0,0) {$\bullet$};
\end{tikzpicture}}
\quad $\xmapsto{\mathscr{D}'}$ \quad
\raisebox{-.6\height}{
\begin{tikzpicture}[scale=0.52]
\def\sep{1.0}
\draw[green!80,thin](0.0,0.0)--(3.0,0.0);
\draw[green!80,thin](3.0,0)--(1.5,-2.598);
\draw[green!80,thin](0.0,0)--(1.5,-2.598);
\draw[green!80,thin](0.5,-0.866)--(2.5,-0.866);
\draw[green!80,thin](2.0,0)--(1.0,-1.7319999999999998);
\draw[green!80,thin](1.0,0)--(2.0,-1.7319999999999998);
\draw[green!80,thin](1.0,-1.732)--(2.0,-1.732);
\draw[green!80,thin](1.0,0)--(0.5,-0.8659999999999999);
\draw[green!80,thin](2.0,0)--(2.5,-0.8659999999999999);
\draw[green!80,thin](1.5,-2.598)--(1.5,-2.598);
\draw[green!80,thin](0.0,0)--(0.0,0.0);
\draw[green!80,thin](3.0,0)--(3.0,0.0);
\draw[green!80,thin](4.0,0.0)--(7.0,0.0);
\draw[green!80,thin](7.0,0)--(5.5,-2.598);
\draw[green!80,thin](4.0,0)--(5.5,-2.598);
\draw[green!80,thin](4.5,-0.866)--(6.5,-0.866);
\draw[green!80,thin](6.0,0)--(5.0,-1.7319999999999998);
\draw[green!80,thin](5.0,0)--(6.0,-1.7319999999999998);
\draw[green!80,thin](5.0,-1.732)--(6.0,-1.732);
\draw[green!80,thin](5.0,0)--(4.5,-0.8659999999999999);
\draw[green!80,thin](6.0,0)--(6.5,-0.8659999999999999);
\draw[green!80,thin](5.5,-2.598)--(5.5,-2.598);
\draw[green!80,thin](4.0,0)--(4.0,0.0);
\draw[green!80,thin](7.0,0)--(7.0,0.0);
\draw[green!80,thin](8.0,0.0)--(11.0,0.0);
\draw[green!80,thin](11.0,0)--(9.5,-2.598);
\draw[green!80,thin](8.0,0)--(9.5,-2.598);
\draw[green!80,thin](8.5,-0.866)--(10.5,-0.866);
\draw[green!80,thin](10.0,0)--(9.0,-1.7319999999999998);
\draw[green!80,thin](9.0,0)--(10.0,-1.7319999999999998);
\draw[green!80,thin](9.0,-1.732)--(10.0,-1.732);
\draw[green!80,thin](9.0,0)--(8.5,-0.8659999999999999);
\draw[green!80,thin](10.0,0)--(10.5,-0.8659999999999999);
\draw[green!80,thin](9.5,-2.598)--(9.5,-2.598);
\draw[green!80,thin](8.0,0)--(8.0,0.0);
\draw[green!80,thin](11.0,0)--(11.0,0.0);
\draw[green!80,thin](12.0,0.0)--(15.0,0.0);
\draw[green!80,thin](15.0,0)--(13.5,-2.598);
\draw[green!80,thin](12.0,0)--(13.5,-2.598);
\draw[green!80,thin](12.5,-0.866)--(14.5,-0.866);
\draw[green!80,thin](14.0,0)--(13.0,-1.7319999999999998);
\draw[green!80,thin](13.0,0)--(14.0,-1.7319999999999998);
\draw[green!80,thin](13.0,-1.732)--(14.0,-1.732);
\draw[green!80,thin](13.0,0)--(12.5,-0.8659999999999999);
\draw[green!80,thin](14.0,0)--(14.5,-0.8659999999999999);
\draw[green!80,thin](13.5,-2.598)--(13.5,-2.598);
\draw[green!80,thin](12.0,0)--(12.0,0.0);
\draw[green!80,thin](15.0,0)--(15.0,0.0);
\node at (0.0,-0.0) {$2$};
\node at (1.0,-0.0) {$1$};
\node at (2.0,-0.0) {$4$};
\node at (3.0,-0.0) {$3$};
\node at (0.5,-0.866) {$2$};
\node at (1.5,-0.866) {$1$};
\node at (2.5,-0.866) {$4$};
\node at (1.0,-1.732) {$2$};
\node at (2.0,-1.732) {$1$};
\node at (1.5,-2.598) {$2$};
\node at (4.0,-0.0) {$2$};
\node at (5.0,-0.0) {$3$};
\node at (6.0,-0.0) {$4$};
\node at (7.0,-0.0) {$2$};
\node at (4.5,-0.866) {$3$};
\node at (5.5,-0.866) {$2$};
\node at (6.5,-0.866) {$3$};
\node at (5.0,-1.732) {$3$};
\node at (6.0,-1.732) {$2$};
\node at (5.5,-2.598) {$3$};
\node at (8.0,-0.0) {$3$};
\node at (9.0,-0.0) {$2$};
\node at (10.0,-0.0) {$4$};
\node at (11.0,-0.0) {$1$};
\node at (8.5,-0.866) {$2$};
\node at (9.5,-0.866) {$4$};
\node at (10.5,-0.866) {$4$};
\node at (9.0,-1.732) {$4$};
\node at (10.0,-1.732) {$4$};
\node at (9.5,-2.598) {$4$};
\node at (12.0,-0.0) {$4$};
\node at (13.0,-0.0) {$1$};
\node at (14.0,-0.0) {$3$};
\node at (15.0,-0.0) {$1$};
\node at (12.5,-0.866) {$1$};
\node at (13.5,-0.866) {$3$};
\node at (14.5,-0.866) {$1$};
\node at (13.0,-1.732) {$3$};
\node at (14.0,-1.732) {$1$};
\node at (13.5,-2.598) {$1$};
\node at (10,-4.2) {};
\end{tikzpicture}}
\caption{The bijection $\mathscr{D}'$ from \Cref{thm:counting-m4}.}
\label{fig:phi-example-m4}
\end{figure}

\begin{theorem}\label{thm:counting-m4}
The map $\mathscr{D}'$ is a bijection $\mc{D}_n\rightarrow\T_{4,n}$.
\end{theorem}

By enumerating the labelings $\mc{D}_n$, we have computed for $n=0,1,2,3$ that $|\T_{4,n}|=1, \: 24, \: 1344,\: 191232$. This last value disagrees with $\frac{1}{5} |\{\text{proper vertex $5$-colorings of } \boxtimes_n\}|$ from \cite[Conjecture~A.5]{Aggarwal-Borodin-Wheeler} which for $n=0,1,2,3$ is equal\footnote{See the OEIS entry A068294.} to $1, \: 24, \: 1344, \: 187008$, disproving the conjecture of Aggarwal--Borodin--Wheeler. However we make a new conjecture for $|\T_{4,n}|$ that has been checked up to $n=7$. This replaces vertex colorings of $\boxtimes_n$ with edge labelings of $\square_n$ and is a direct extension of the equinumerosity of $\mc{P}_n$ and $\T_{3,n}$.


\begin{conj}\label{conj:counting-m4-proper-labeling}
$|\T_{4,n}|$ equals the number of edge labelings of $\square_n$ with four labels such that the four sides of each face have distinct labels. 
\end{conj}

\begin{comment}
\section{Schubert structure constants}
\label{sec:geometry}
In this section we derive the geometric results stated in \Cref{sec:intro-geometry}.

\subsection{Puzzle conversion}
\label{sec:puzzle-conversion}
A \emph{$0/1/10$-puzzle} is a labeling of the edges of $\Delta_n$ with labels $0$, $1$, and $10$ so that each $\Delta$-oriented face is labeled in one of the following ways

\begin{center}
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture},
\end{center}
and so that each $\nabla$-oriented face is labeled by a $180^{\circ}$ rotation of one of these\footnote{See \cite[Section 5]{Knutson-Zinn-Justin-3} for the relationship between these puzzles and others which have appeared in the literature.}. The boundary conditions of such a puzzle are the $0,1$-strings $\xi^{(1)},\xi^{(2)},\xi^{(3)}$ obtained by reading the labels of the boundary edges of the three sides of $\Delta_n$ clockwise from the basepoint. We write $\tilde{\mc{P}}_n(\bs{\xi})$ for the set of $0/1/10$-puzzles on $\Delta_n$ with boundary conditions $\bs{\xi}$ and $\tilde{\mc{P}}_n$ for the set of all $0/1/10$-puzzles on $\Delta_n$. 

We are grateful to Allen Knutson for sharing with us the following correspondence between $1/2/3$-puzzles and $0/1/10$-puzzles. Given a boundary condition $\bs{\lambda}=(\lambda^{(1)},\lambda^{(2)},\lambda^{(3)})$, let $\partn(\bs{\lambda})$ be the $0,1$-strings $(\xi^{(1)},\xi^{(2)},\xi^{(3)})$ obtained by applying the transformation:

%\begin{figure}
    \begin{center}
    \begin{tikzpicture}[scale=1]
    \draw[blue!80,thin] (0,0) -- ({sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (1,0) -- ({1+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$3$}};
    \draw[blue!80,thin] (2,0) -- ({2+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (6.5,0.42) -- ({6.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$3$}};
    \draw[blue!80,thin] (8,.42) -- ({8+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (9.5,.42) -- ({9.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (3.5,1) -- ({3.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (4.5,1) -- ({4.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (5.5,1) -- ({5.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$3$}};
\end{tikzpicture} 
\\ $\updownarrow$ \\
\begin{tikzpicture}[scale=1]
    \draw[cyan!90,thin] (0,0) -- ({sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (1,0) -- ({1+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (2,0) -- ({2+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (6.5,0.42) -- ({6.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (8,.42) -- ({8+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (9.5,.42) -- ({9.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (3.5,1) -- ({3.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (4.5,1) -- ({4.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (5.5,1) -- ({5.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture}.
\end{center}
%    \caption{The transformation between $1/2/3$-puzzle edge labels (top) and $0/1/10$-puzzle edge labels (bottom).}
%    \label{fig:puzzle-cryptomorphism}
%\end{figure}
We write $\toprow$ for the inverse to $\partn$. We have following proposition.

\begin{prop}
\label{prop:puzzle-cryptomorphism}
For any boundary condition $\bs{\lambda}$, the transformation of edge labels shown above determines a bijection $\mc{P}_n(\bs{\lambda}) \to \tilde{\mc{P}}_n(\partn(\bs{\lambda}))$.
\end{prop}

We write $\tilde{P}$ for the $0/1/10$-puzzle corresponding to a $1/2/3$-puzzle $P$.

\subsection{The splitting lemma} 
\label{sec:splitting}
We prove an important lemma which will allow us to reduce to the case $m=3$.

\begin{defin}
\label{def:dual-array}
Let $\mu \in \{a,b\}^n$ be a string on two letters. The \emph{dual string} $\mu^\dagger$ is obtained by reversing $\mu$ and swapping $a$'s with $b$'s. If $T^{(i)}$ is a triangle such that $T^{(i)}_{j,k} \in \{a,b\}$ for all $1\leq j \leq k \leq n$, the \emph{dual triangle} $(T^{(i)})^{\dagger}$ is obtained by dualizing each row of $T^{(i)}$.
\end{defin}

\begin{defin}
\label{def:split-map}
Fix $m\geq3$ and $\d=(d_1\leq d_2\leq\cdots\leq d_{m-1}\leq n)$ and for $i=1,\ldots,m$ fix Schubert strings $\lambda^{(i)}$ of type
\[
(m-i)^{d_{m-i}} \cdot m^{d_{m-i+1}-d_{m-i}} \cdot (m-i+1)^{n-d_{m-i+1}}.
\]
We define a map $\splt$ sending each $T \in \T_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$ to a pair $\splt(T)=(R,S)$ of arrays, as follows. For each $k=1,\ldots,n$, let $a_k$ be the unique element of the difference of multisets
\[
\left\{(m-2)^k,(m-1)^k,m^k\right\} \setminus \left(T^{(1)}_{\bullet,k} \cup T^{(2)}_{\bullet,k} \cup R^{(3)}_{\bullet,k-1}\right),
\]
where $R^{(3)}_{\bullet,k}$ is defined inductively by $R^{(3)}_{j,k}=R^{(3)}_{j-1,k-1}$ for $j=2,\ldots,k$ and $R^{(3)}_{1,k}=a_k$, with the base case $R^{(3)}_{\bullet,0}=\emptyset$. Then define $R$ and $S$ by
\begin{align*}
    R &= (T^{(1)},T^{(2)},R^{(3)}), \\
    S &= ((R^{(3)})^{\dagger}, T^{(3)}, \ldots, T^{(m)}).
\end{align*}
\end{defin}

\begin{ex}
\label{ex:split}
We give an example of the map $\splt$. Let $m=4$, $n=5$ and $\alpha(\bs{d})=(1,2,0,2)$. Take $\lambda^{(1)}=33443$, $\lambda^{(2)}=22323$, $\lambda^{(3)}=12424$, and $\lambda^{(4)}=11411$. Let $T\in\mc{T}_{4,5}(\lambda^{(1)},\ldots,\lambda^{(4)})$ be as shown below (top). Then $\splt(T)=(R,S) \in \T_{3,5} \times \T_{3,5}$ is shown on bottom. Note that $T=(R^{(1)},R^{(2)},S^{(2)},S^{(3)})$ and that $(R^{(3)})^\dagger = S^{(1)}$.
%Correspondingly, $w^{(1)}=12534$, $w^{(2)}=12435$, $w^{(3)}=13524$, $w^{(4)}=31245$, with their length summing up to $8=\binom{5}{2}-\sum_{i=1}^4\binom{\alpha_i}{2}$.
%\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}[scale=0.500000000000000]
\def\sep{2}
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](6.00000000000000,0.000000000000000)--(10.0000000000000,0.000000000000000);
\draw[green!80,thin](10,0)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6,0)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6.50000000000000,-0.866000000000000)--(9.50000000000000,-0.866000000000000);
\draw[green!80,thin](9,0)--(7.50000000000000,-2.59800000000000);
\draw[green!80,thin](7,0)--(8.50000000000000,-2.59800000000000);
\draw[green!80,thin](7.00000000000000,-1.73200000000000)--(9.00000000000000,-1.73200000000000);
\draw[green!80,thin](8,0)--(7.00000000000000,-1.73200000000000);
\draw[green!80,thin](8,0)--(9.00000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,-2.59800000000000)--(8.50000000000000,-2.59800000000000);
\draw[green!80,thin](7,0)--(6.50000000000000,-0.866000000000000);
\draw[green!80,thin](9,0)--(9.50000000000000,-0.866000000000000);
\draw[green!80,thin](8.00000000000000,-3.46400000000000)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6,0)--(6.00000000000000,0.000000000000000);
\draw[green!80,thin](10,0)--(10.0000000000000,0.000000000000000);
\draw[green!80,thin](12.0000000000000,0.000000000000000)--(16.0000000000000,0.000000000000000);
\draw[green!80,thin](16,0)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12,0)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12.5000000000000,-0.866000000000000)--(15.5000000000000,-0.866000000000000);
\draw[green!80,thin](15,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](13,0)--(14.5000000000000,-2.59800000000000);
\draw[green!80,thin](13.0000000000000,-1.73200000000000)--(15.0000000000000,-1.73200000000000);
\draw[green!80,thin](14,0)--(13.0000000000000,-1.73200000000000);
\draw[green!80,thin](14,0)--(15.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.5000000000000,-2.59800000000000)--(14.5000000000000,-2.59800000000000);
\draw[green!80,thin](13,0)--(12.5000000000000,-0.866000000000000);
\draw[green!80,thin](15,0)--(15.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,-3.46400000000000)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12,0)--(12.0000000000000,0.000000000000000);
\draw[green!80,thin](16,0)--(16.0000000000000,0.000000000000000);
\draw[green!80,thin](18.0000000000000,0.000000000000000)--(22.0000000000000,0.000000000000000);
\draw[green!80,thin](22,0)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18,0)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18.5000000000000,-0.866000000000000)--(21.5000000000000,-0.866000000000000);
\draw[green!80,thin](21,0)--(19.5000000000000,-2.59800000000000);
\draw[green!80,thin](19,0)--(20.5000000000000,-2.59800000000000);
\draw[green!80,thin](19.0000000000000,-1.73200000000000)--(21.0000000000000,-1.73200000000000);
\draw[green!80,thin](20,0)--(19.0000000000000,-1.73200000000000);
\draw[green!80,thin](20,0)--(21.0000000000000,-1.73200000000000);
\draw[green!80,thin](19.5000000000000,-2.59800000000000)--(20.5000000000000,-2.59800000000000);
\draw[green!80,thin](19,0)--(18.5000000000000,-0.866000000000000);
\draw[green!80,thin](21,0)--(21.5000000000000,-0.866000000000000);
\draw[green!80,thin](20.0000000000000,-3.46400000000000)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18,0)--(18.0000000000000,0.000000000000000);
\draw[green!80,thin](22,0)--(22.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$3$};
\node at (1.00000000000000,-0.000000000000000) {$3$};
\node at (2.00000000000000,-0.000000000000000) {$4$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$3$};
\node at (0.500000000000000,-0.866000000000000) {$3$};
\node at (1.50000000000000,-0.866000000000000) {$3$};
\node at (2.50000000000000,-0.866000000000000) {$4$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$3$};
\node at (2.00000000000000,-1.73200000000000) {$3$};
\node at (3.00000000000000,-1.73200000000000) {$4$};
\node at (1.50000000000000,-2.59800000000000) {$3$};
\node at (2.50000000000000,-2.59800000000000) {$3$};
\node at (2.00000000000000,-3.46400000000000) {$3$};
\node at (6.00000000000000,-0.000000000000000) {$2$};
\node at (7.00000000000000,-0.000000000000000) {$2$};
\node at (8.00000000000000,-0.000000000000000) {$3$};
\node at (9.00000000000000,-0.000000000000000) {$2$};
\node at (10.0000000000000,-0.000000000000000) {$3$};
\node at (6.50000000000000,-0.866000000000000) {$2$};
\node at (7.50000000000000,-0.866000000000000) {$3$};
\node at (8.50000000000000,-0.866000000000000) {$2$};
\node at (9.50000000000000,-0.866000000000000) {$3$};
\node at (7.00000000000000,-1.73200000000000) {$2$};
\node at (8.00000000000000,-1.73200000000000) {$3$};
\node at (9.00000000000000,-1.73200000000000) {$2$};
\node at (7.50000000000000,-2.59800000000000) {$2$};
\node at (8.50000000000000,-2.59800000000000) {$4$};
\node at (8.00000000000000,-3.46400000000000) {$2$};
\node at (12.0000000000000,-0.000000000000000) {$1$};
\node at (13.0000000000000,-0.000000000000000) {$2$};
\node at (14.0000000000000,-0.000000000000000) {$4$};
\node at (15.0000000000000,-0.000000000000000) {$2$};
\node at (16.0000000000000,-0.000000000000000) {$4$};
\node at (12.5000000000000,-0.866000000000000) {$2$};
\node at (13.5000000000000,-0.866000000000000) {$1$};
\node at (14.5000000000000,-0.866000000000000) {$2$};
\node at (15.5000000000000,-0.866000000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$4$};
\node at (14.0000000000000,-1.73200000000000) {$2$};
\node at (15.0000000000000,-1.73200000000000) {$1$};
\node at (13.5000000000000,-2.59800000000000) {$2$};
\node at (14.5000000000000,-2.59800000000000) {$4$};
\node at (14.0000000000000,-3.46400000000000) {$4$};
\node at (18.0000000000000,-0.000000000000000) {$1$};
\node at (19.0000000000000,-0.000000000000000) {$1$};
\node at (20.0000000000000,-0.000000000000000) {$4$};
\node at (21.0000000000000,-0.000000000000000) {$1$};
\node at (22.0000000000000,-0.000000000000000) {$1$};
\node at (18.5000000000000,-0.866000000000000) {$1$};
\node at (19.5000000000000,-0.866000000000000) {$4$};
\node at (20.5000000000000,-0.866000000000000) {$1$};
\node at (21.5000000000000,-0.866000000000000) {$1$};
\node at (19.0000000000000,-1.73200000000000) {$4$};
\node at (20.0000000000000,-1.73200000000000) {$1$};
\node at (21.0000000000000,-1.73200000000000) {$1$};
\node at (19.5000000000000,-2.59800000000000) {$1$};
\node at (20.5000000000000,-2.59800000000000) {$1$};
\node at (20.0000000000000,-3.46400000000000) {$1$};
\draw[black,thin](-0.5,0.5)--(22.5,0.5)--(22.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}

\


\begin{tikzpicture}[scale=0.350000000000000]
\def\sep{1.50000000000000};
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](5.50000000000000,0.000000000000000)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](6.00000000000000,-0.866000000000000)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(7.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,-1.73200000000000)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.00000000000000,-2.59800000000000)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(6.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](7.50000000000000,-3.46400000000000)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(5.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](11.0000000000000,0.000000000000000)--(15.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.5000000000000,-0.866000000000000)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(12.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,-1.73200000000000)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(12.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](12.5000000000000,-2.59800000000000)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](13.0000000000000,-3.46400000000000)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(11.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(15.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$3$};
\node at (1.00000000000000,-0.000000000000000) {$3$};
\node at (2.00000000000000,-0.000000000000000) {$4$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$3$};
\node at (0.500000000000000,-0.866000000000000) {$3$};
\node at (1.50000000000000,-0.866000000000000) {$3$};
\node at (2.50000000000000,-0.866000000000000) {$4$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$3$};
\node at (2.00000000000000,-1.73200000000000) {$3$};
\node at (3.00000000000000,-1.73200000000000) {$4$};
\node at (1.50000000000000,-2.59800000000000) {$3$};
\node at (2.50000000000000,-2.59800000000000) {$3$};
\node at (2.00000000000000,-3.46400000000000) {$3$};
\node at (5.50000000000000,-0.000000000000000) {$2$};
\node at (6.50000000000000,-0.000000000000000) {$2$};
\node at (7.50000000000000,-0.000000000000000) {$3$};
\node at (8.50000000000000,-0.000000000000000) {$2$};
\node at (9.50000000000000,-0.000000000000000) {$3$};
\node at (6.00000000000000,-0.866000000000000) {$2$};
\node at (7.00000000000000,-0.866000000000000) {$3$};
\node at (8.00000000000000,-0.866000000000000) {$2$};
\node at (9.00000000000000,-0.866000000000000) {$3$};
\node at (6.50000000000000,-1.73200000000000) {$2$};
\node at (7.50000000000000,-1.73200000000000) {$3$};
\node at (8.50000000000000,-1.73200000000000) {$2$};
\node at (7.00000000000000,-2.59800000000000) {$2$};
\node at (8.00000000000000,-2.59800000000000) {$4$};
\node at (7.50000000000000,-3.46400000000000) {$2$};
\node at (11.0000000000000,-0.000000000000000) {$4$};
\node at (12.0000000000000,-0.000000000000000) {$2$};
\node at (13.0000000000000,-0.000000000000000) {$4$};
\node at (14.0000000000000,-0.000000000000000) {$2$};
\node at (15.0000000000000,-0.000000000000000) {$4$};
\node at (11.5000000000000,-0.866000000000000) {$2$};
\node at (12.5000000000000,-0.866000000000000) {$4$};
\node at (13.5000000000000,-0.866000000000000) {$2$};
\node at (14.5000000000000,-0.866000000000000) {$4$};
\node at (12.0000000000000,-1.73200000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$2$};
\node at (14.0000000000000,-1.73200000000000) {$4$};
\node at (12.5000000000000,-2.59800000000000) {$2$};
\node at (13.5000000000000,-2.59800000000000) {$4$};
\node at (13.0000000000000,-3.46400000000000) {$4$};
\draw[black,thin] (-0.5,0.5)--(15.5,0.5)--(15.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.350000000000000]
\def\sep{1.50000000000000};
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](5.50000000000000,0.000000000000000)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](6.00000000000000,-0.866000000000000)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(7.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,-1.73200000000000)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.00000000000000,-2.59800000000000)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(6.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](7.50000000000000,-3.46400000000000)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(5.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](11.0000000000000,0.000000000000000)--(15.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.5000000000000,-0.866000000000000)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(12.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,-1.73200000000000)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(12.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](12.5000000000000,-2.59800000000000)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](13.0000000000000,-3.46400000000000)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(11.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(15.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$2$};
\node at (1.00000000000000,-0.000000000000000) {$4$};
\node at (2.00000000000000,-0.000000000000000) {$2$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$2$};
\node at (0.500000000000000,-0.866000000000000) {$2$};
\node at (1.50000000000000,-0.866000000000000) {$4$};
\node at (2.50000000000000,-0.866000000000000) {$2$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$2$};
\node at (2.00000000000000,-1.73200000000000) {$4$};
\node at (3.00000000000000,-1.73200000000000) {$2$};
\node at (1.50000000000000,-2.59800000000000) {$2$};
\node at (2.50000000000000,-2.59800000000000) {$4$};
\node at (2.00000000000000,-3.46400000000000) {$2$};
\node at (5.50000000000000,-0.000000000000000) {$1$};
\node at (6.50000000000000,-0.000000000000000) {$2$};
\node at (7.50000000000000,-0.000000000000000) {$4$};
\node at (8.50000000000000,-0.000000000000000) {$2$};
\node at (9.50000000000000,-0.000000000000000) {$4$};
\node at (6.00000000000000,-0.866000000000000) {$2$};
\node at (7.00000000000000,-0.866000000000000) {$1$};
\node at (8.00000000000000,-0.866000000000000) {$2$};
\node at (9.00000000000000,-0.866000000000000) {$4$};
\node at (6.50000000000000,-1.73200000000000) {$4$};
\node at (7.50000000000000,-1.73200000000000) {$2$};
\node at (8.50000000000000,-1.73200000000000) {$1$};
\node at (7.00000000000000,-2.59800000000000) {$2$};
\node at (8.00000000000000,-2.59800000000000) {$4$};
\node at (7.50000000000000,-3.46400000000000) {$4$};
\node at (11.0000000000000,-0.000000000000000) {$1$};
\node at (12.0000000000000,-0.000000000000000) {$1$};
\node at (13.0000000000000,-0.000000000000000) {$4$};
\node at (14.0000000000000,-0.000000000000000) {$1$};
\node at (15.0000000000000,-0.000000000000000) {$1$};
\node at (11.5000000000000,-0.866000000000000) {$1$};
\node at (12.5000000000000,-0.866000000000000) {$4$};
\node at (13.5000000000000,-0.866000000000000) {$1$};
\node at (14.5000000000000,-0.866000000000000) {$1$};
\node at (12.0000000000000,-1.73200000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$1$};
\node at (14.0000000000000,-1.73200000000000) {$1$};
\node at (12.5000000000000,-2.59800000000000) {$1$};
\node at (13.5000000000000,-2.59800000000000) {$1$};
\node at (13.0000000000000,-3.46400000000000) {$1$};
\draw[black,thin] (-0.5,0.5)--(15.5,0.5)--(15.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}
%\caption{The interlacing triangular array $T \in \T_{4,5}$ (top) and the arrays $\splt(T)=(R,S) \in \T_{3,5} \times \T_{3,5}$ (bottom).}
%\label{fig:proof-merge-example}
%\end{figure}
\end{center}
\end{ex}


%\begin{defin} We say that an interlacing triangular array $T$ of rank $m$ and height $n$ is of \emph{Schubert type} $\d=(d_1\leq\cdots\leq d_{m-1})$ if the top row $T^{(i)}_n$ is a Schubert string of type $(m-i)^{d_{m-i}}m^{d_{m-i+1}-d_{m-i}}(m-i+1)^{n-d_{m-i+1}}$ for $i=1,\ldots,m$. By convention, $d_0=0$ and $d_m=n$. Denote the set of all such interlacing triangular arrays as $\mc{T}_{m,n}^{\d}$. \end{defin}

Given any totally ordered set $\Sigma$ of cardinality $m$, we write $\T_{m,n}^{\Sigma}(\bs{\lambda})$ for the set of interlacing triangular arrays of rank $m$, height $n$, and top row $\bs{\lambda}$, with entries from $\Sigma$ (instead of from $[m]$); clearly the choice of $\Sigma$ does not affect the cardinality of this set. We can now state the splitting lemma.

\begin{lemma}\label{lem:splitting-interlacing}
Fix $m\geq3$ and $\d=(d_1\leq d_2\leq\cdots\leq d_{m-1}\leq n)$ and for $i=1,\ldots,m$, fix Schubert strings $\lambda^{(i)}$ of type $(m-i)^{d_{m-i}}m^{d_{m-i+1}-d_{m-i}}(m-i+1)^{n-d_{m-i+1}}$. Then $\splt$ is a bijection from $\T_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$ to
\begin{equation}
\label{eq:splitting-union}
\bigsqcup_{\mu}\T_{3,n}^{\{m-2,m-1,m\}}(\lambda^{(1)},\lambda^{(2)},\mu)\times\T_{m-1,n}^{[m]\setminus \{m-1\}}(\mu^{\dagger},\lambda^{(3)},\ldots,\lambda^{(m)}),
\end{equation}
where the union runs over $\mu$ of type $(m-2)^{n-d_{m-2}}m^{d_{m-2}}$.
\end{lemma}

\subsection{Avoiding puzzle pieces} The geometric interpretations of $0/1/10$-puzzles in terms of Schubert classes in cohomology, the $G_{\xi}$ basis in $K$-theory, and the $G^{\ast}_{\xi}$ basis in $K$-theory involve forbidding the $\Delta$- and $\nabla$-oriented $10$-$10$-$10$ pieces, or one or the other of these pieces. We call these two pieces, as well as their $1/2/3$-analogs, the \emph{$K$-pieces}. We now describe the forbidden substructures in interlacing triangular arrays corresponding to forbidden one or both of the $K$-pieces.

\begin{prop}
\label{prop:avoiding-delta-10-10-10}
Suppose that $\lambda^{(1)} \in \{2,3\}^n, \lambda^{(2)} \in \{1,2,3\}^n,$ and $\lambda^{(3)} \in \{1,3\}^n$. Then the following are equivalent:
\begin{itemize}[leftmargin=0.3in]
    \item[(a)] The $0/1/10$-puzzle $\tilde{P}$ with boundary conditions $\partn(\bs{\lambda})$ avoids the piece 
    \raisebox{-0.5\height}{
    \begin{tikzpicture}[scale=0.8, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\textcolor{black}{$10$}};
\end{tikzpicture}}.
    \item[(b)] The $1/2/3$-puzzle $P$ with boundary conditions $\bs{\lambda}$ avoids the piece 
    \raisebox{-0.5\height}{
    \begin{tikzpicture}[scale=0.8, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[blue!80,thin] (c0) -- (d0) node [midway] {\textcolor{black}{$1$}};
    \draw[blue!80,thin] (c0) -- (d1) node [midway] {\textcolor{black}{$2$}};
    \draw[blue!80,thin] (d0) -- (d1) node [midway] {\textcolor{black}{$3$}};
\end{tikzpicture}}.
    \item[(c)] The array $T=\mathscr{T}(P)$ with top row $\bs{\lambda}$ avoids \raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.8]
    \draw[green!80,thin] (0,0)--(-0.5,.866);
    \node at (-0.5,.866) {$3$};
    \node at (0,0) {$1$};
\end{tikzpicture}} and 
\raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.8]
    \draw[green!80,thin] (0,0)--(-0.5,.866)--(0.5,.866)--(0,0);
    \node at (-0.5,.866) {$3$};
    \node at (0.5,.866) {$2$};
    \node at (0,0) {$3$};
\end{tikzpicture}} in $T^{(2)}$.
  \item[(d)] The array $T=\mathscr{T}(P)$ with top row $\bs{\lambda}$ avoids \raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.8]
    \draw[green!80,thin] (0,0)--(0.5,.866);
    \node at (0.5,.866) {$3$};
    \node at (0,0) {$2$};
\end{tikzpicture}} and 
\raisebox{-0.5\height}{
\begin{tikzpicture}[scale=0.8]
    \draw[green!80,thin] (0,0)--(-0.5,.866)--(0.5,.866)--(0,0);
    \node at (-0.5,.866) {$1$};
    \node at (0.5,.866) {$3$};
    \node at (0,0) {$3$};
\end{tikzpicture}} in $T^{(2)}$.
\end{itemize}
\end{prop}

There is a closely analogous proposition for the other $10$-$10$-$10$ piece.

%\subsection{Proofs of geometric interpretations}
\end{comment}

\section{Ingredients in the geometric interpretations}
\label{sec:geometric-sketch}

We end by giving a couple of combinatorial ingredients used in our proofs of \Cref{thm:k-theory,thm:dual-k,thm:2-step,thm:ssm}. The full details are available in \cite[Section 5]{arxiv-version}.

\subsection{The splitting lemma}
\begin{lemma}\label{lem:splitting-interlacing}
Fix $m\geq3$ and $\d$ and strings $\lambda^{(i)}$ of type $(m-i)^{d_{m-i}}m^{d_{m-i+1}-d_{m-i}}(m-i+1)^{n-d_{m-i+1}}$. Then there is a bijection $\splt$ from $\T_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$ to
\begin{equation}
\label{eq:splitting-union}
\bigsqcup_{\mu}\T_{3,n}(\lambda^{(1)},\lambda^{(2)},\mu)\times\T_{m-1,n}(\mu^{\dagger},\lambda^{(3)},\ldots,\lambda^{(m)}),
\end{equation}
where $\mu$ runs over type $(m-2)^{n-d_{m-2}}m^{d_{m-2}}$ and $\mu^{\dagger}$ reverses and swaps the letters of $\mu$.
\end{lemma}

\begin{ex}
\label{ex:split}
Let $T\in\mc{T}_{4,5}$ be as below; then $\splt(T) \in \T_{3,5} \times \T_{3,5}$ is shown on bottom. 
%Correspondingly, $w^{(1)}=12534$, $w^{(2)}=12435$, $w^{(3)}=13524$, $w^{(4)}=31245$, with their length summing up to $8=\binom{5}{2}-\sum_{i=1}^4\binom{\alpha_i}{2}$.
%\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}[scale=0.500000000000000]
\def\sep{2}
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](6.00000000000000,0.000000000000000)--(10.0000000000000,0.000000000000000);
\draw[green!80,thin](10,0)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6,0)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6.50000000000000,-0.866000000000000)--(9.50000000000000,-0.866000000000000);
\draw[green!80,thin](9,0)--(7.50000000000000,-2.59800000000000);
\draw[green!80,thin](7,0)--(8.50000000000000,-2.59800000000000);
\draw[green!80,thin](7.00000000000000,-1.73200000000000)--(9.00000000000000,-1.73200000000000);
\draw[green!80,thin](8,0)--(7.00000000000000,-1.73200000000000);
\draw[green!80,thin](8,0)--(9.00000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,-2.59800000000000)--(8.50000000000000,-2.59800000000000);
\draw[green!80,thin](7,0)--(6.50000000000000,-0.866000000000000);
\draw[green!80,thin](9,0)--(9.50000000000000,-0.866000000000000);
\draw[green!80,thin](8.00000000000000,-3.46400000000000)--(8.00000000000000,-3.46400000000000);
\draw[green!80,thin](6,0)--(6.00000000000000,0.000000000000000);
\draw[green!80,thin](10,0)--(10.0000000000000,0.000000000000000);
\draw[green!80,thin](12.0000000000000,0.000000000000000)--(16.0000000000000,0.000000000000000);
\draw[green!80,thin](16,0)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12,0)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12.5000000000000,-0.866000000000000)--(15.5000000000000,-0.866000000000000);
\draw[green!80,thin](15,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](13,0)--(14.5000000000000,-2.59800000000000);
\draw[green!80,thin](13.0000000000000,-1.73200000000000)--(15.0000000000000,-1.73200000000000);
\draw[green!80,thin](14,0)--(13.0000000000000,-1.73200000000000);
\draw[green!80,thin](14,0)--(15.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.5000000000000,-2.59800000000000)--(14.5000000000000,-2.59800000000000);
\draw[green!80,thin](13,0)--(12.5000000000000,-0.866000000000000);
\draw[green!80,thin](15,0)--(15.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,-3.46400000000000)--(14.0000000000000,-3.46400000000000);
\draw[green!80,thin](12,0)--(12.0000000000000,0.000000000000000);
\draw[green!80,thin](16,0)--(16.0000000000000,0.000000000000000);
\draw[green!80,thin](18.0000000000000,0.000000000000000)--(22.0000000000000,0.000000000000000);
\draw[green!80,thin](22,0)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18,0)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18.5000000000000,-0.866000000000000)--(21.5000000000000,-0.866000000000000);
\draw[green!80,thin](21,0)--(19.5000000000000,-2.59800000000000);
\draw[green!80,thin](19,0)--(20.5000000000000,-2.59800000000000);
\draw[green!80,thin](19.0000000000000,-1.73200000000000)--(21.0000000000000,-1.73200000000000);
\draw[green!80,thin](20,0)--(19.0000000000000,-1.73200000000000);
\draw[green!80,thin](20,0)--(21.0000000000000,-1.73200000000000);
\draw[green!80,thin](19.5000000000000,-2.59800000000000)--(20.5000000000000,-2.59800000000000);
\draw[green!80,thin](19,0)--(18.5000000000000,-0.866000000000000);
\draw[green!80,thin](21,0)--(21.5000000000000,-0.866000000000000);
\draw[green!80,thin](20.0000000000000,-3.46400000000000)--(20.0000000000000,-3.46400000000000);
\draw[green!80,thin](18,0)--(18.0000000000000,0.000000000000000);
\draw[green!80,thin](22,0)--(22.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$3$};
\node at (1.00000000000000,-0.000000000000000) {$3$};
\node at (2.00000000000000,-0.000000000000000) {$4$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$3$};
\node at (0.500000000000000,-0.866000000000000) {$3$};
\node at (1.50000000000000,-0.866000000000000) {$3$};
\node at (2.50000000000000,-0.866000000000000) {$4$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$3$};
\node at (2.00000000000000,-1.73200000000000) {$3$};
\node at (3.00000000000000,-1.73200000000000) {$4$};
\node at (1.50000000000000,-2.59800000000000) {$3$};
\node at (2.50000000000000,-2.59800000000000) {$3$};
\node at (2.00000000000000,-3.46400000000000) {$3$};
\node at (6.00000000000000,-0.000000000000000) {$2$};
\node at (7.00000000000000,-0.000000000000000) {$2$};
\node at (8.00000000000000,-0.000000000000000) {$3$};
\node at (9.00000000000000,-0.000000000000000) {$2$};
\node at (10.0000000000000,-0.000000000000000) {$3$};
\node at (6.50000000000000,-0.866000000000000) {$2$};
\node at (7.50000000000000,-0.866000000000000) {$3$};
\node at (8.50000000000000,-0.866000000000000) {$2$};
\node at (9.50000000000000,-0.866000000000000) {$3$};
\node at (7.00000000000000,-1.73200000000000) {$2$};
\node at (8.00000000000000,-1.73200000000000) {$3$};
\node at (9.00000000000000,-1.73200000000000) {$2$};
\node at (7.50000000000000,-2.59800000000000) {$2$};
\node at (8.50000000000000,-2.59800000000000) {$4$};
\node at (8.00000000000000,-3.46400000000000) {$2$};
\node at (12.0000000000000,-0.000000000000000) {$1$};
\node at (13.0000000000000,-0.000000000000000) {$2$};
\node at (14.0000000000000,-0.000000000000000) {$4$};
\node at (15.0000000000000,-0.000000000000000) {$2$};
\node at (16.0000000000000,-0.000000000000000) {$4$};
\node at (12.5000000000000,-0.866000000000000) {$2$};
\node at (13.5000000000000,-0.866000000000000) {$1$};
\node at (14.5000000000000,-0.866000000000000) {$2$};
\node at (15.5000000000000,-0.866000000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$4$};
\node at (14.0000000000000,-1.73200000000000) {$2$};
\node at (15.0000000000000,-1.73200000000000) {$1$};
\node at (13.5000000000000,-2.59800000000000) {$2$};
\node at (14.5000000000000,-2.59800000000000) {$4$};
\node at (14.0000000000000,-3.46400000000000) {$4$};
\node at (18.0000000000000,-0.000000000000000) {$1$};
\node at (19.0000000000000,-0.000000000000000) {$1$};
\node at (20.0000000000000,-0.000000000000000) {$4$};
\node at (21.0000000000000,-0.000000000000000) {$1$};
\node at (22.0000000000000,-0.000000000000000) {$1$};
\node at (18.5000000000000,-0.866000000000000) {$1$};
\node at (19.5000000000000,-0.866000000000000) {$4$};
\node at (20.5000000000000,-0.866000000000000) {$1$};
\node at (21.5000000000000,-0.866000000000000) {$1$};
\node at (19.0000000000000,-1.73200000000000) {$4$};
\node at (20.0000000000000,-1.73200000000000) {$1$};
\node at (21.0000000000000,-1.73200000000000) {$1$};
\node at (19.5000000000000,-2.59800000000000) {$1$};
\node at (20.5000000000000,-2.59800000000000) {$1$};
\node at (20.0000000000000,-3.46400000000000) {$1$};
\draw[black,thin](-0.5,0.5)--(22.5,0.5)--(22.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}

\


\begin{tikzpicture}[scale=0.350000000000000]
\def\sep{1.50000000000000}
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](5.50000000000000,0.000000000000000)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](6.00000000000000,-0.866000000000000)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(7.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,-1.73200000000000)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.00000000000000,-2.59800000000000)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(6.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](7.50000000000000,-3.46400000000000)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(5.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](11.0000000000000,0.000000000000000)--(15.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.5000000000000,-0.866000000000000)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(12.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,-1.73200000000000)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(12.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](12.5000000000000,-2.59800000000000)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](13.0000000000000,-3.46400000000000)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(11.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(15.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$3$};
\node at (1.00000000000000,-0.000000000000000) {$3$};
\node at (2.00000000000000,-0.000000000000000) {$4$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$3$};
\node at (0.500000000000000,-0.866000000000000) {$3$};
\node at (1.50000000000000,-0.866000000000000) {$3$};
\node at (2.50000000000000,-0.866000000000000) {$4$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$3$};
\node at (2.00000000000000,-1.73200000000000) {$3$};
\node at (3.00000000000000,-1.73200000000000) {$4$};
\node at (1.50000000000000,-2.59800000000000) {$3$};
\node at (2.50000000000000,-2.59800000000000) {$3$};
\node at (2.00000000000000,-3.46400000000000) {$3$};
\node at (5.50000000000000,-0.000000000000000) {$2$};
\node at (6.50000000000000,-0.000000000000000) {$2$};
\node at (7.50000000000000,-0.000000000000000) {$3$};
\node at (8.50000000000000,-0.000000000000000) {$2$};
\node at (9.50000000000000,-0.000000000000000) {$3$};
\node at (6.00000000000000,-0.866000000000000) {$2$};
\node at (7.00000000000000,-0.866000000000000) {$3$};
\node at (8.00000000000000,-0.866000000000000) {$2$};
\node at (9.00000000000000,-0.866000000000000) {$3$};
\node at (6.50000000000000,-1.73200000000000) {$2$};
\node at (7.50000000000000,-1.73200000000000) {$3$};
\node at (8.50000000000000,-1.73200000000000) {$2$};
\node at (7.00000000000000,-2.59800000000000) {$2$};
\node at (8.00000000000000,-2.59800000000000) {$4$};
\node at (7.50000000000000,-3.46400000000000) {$2$};
\node at (11.0000000000000,-0.000000000000000) {$4$};
\node at (12.0000000000000,-0.000000000000000) {$2$};
\node at (13.0000000000000,-0.000000000000000) {$4$};
\node at (14.0000000000000,-0.000000000000000) {$2$};
\node at (15.0000000000000,-0.000000000000000) {$4$};
\node at (11.5000000000000,-0.866000000000000) {$2$};
\node at (12.5000000000000,-0.866000000000000) {$4$};
\node at (13.5000000000000,-0.866000000000000) {$2$};
\node at (14.5000000000000,-0.866000000000000) {$4$};
\node at (12.0000000000000,-1.73200000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$2$};
\node at (14.0000000000000,-1.73200000000000) {$4$};
\node at (12.5000000000000,-2.59800000000000) {$2$};
\node at (13.5000000000000,-2.59800000000000) {$4$};
\node at (13.0000000000000,-3.46400000000000) {$4$};
\draw[black,thin] (-0.5,0.5)--(15.5,0.5)--(15.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.350000000000000]
\def\sep{1.50000000000000}
\draw[green!80,thin](0.000000000000000,0.000000000000000)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.500000000000000,-0.866000000000000)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(1.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,-1.73200000000000)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(1.00000000000000,-1.73200000000000);
\draw[green!80,thin](2.00000000000000,0)--(3.00000000000000,-1.73200000000000);
\draw[green!80,thin](1.50000000000000,-2.59800000000000)--(2.50000000000000,-2.59800000000000);
\draw[green!80,thin](1.00000000000000,0)--(0.500000000000000,-0.866000000000000);
\draw[green!80,thin](3.00000000000000,0)--(3.50000000000000,-0.866000000000000);
\draw[green!80,thin](2.00000000000000,-3.46400000000000)--(2.00000000000000,-3.46400000000000);
\draw[green!80,thin](0.000000000000000,0)--(0.000000000000000,0.000000000000000);
\draw[green!80,thin](4.00000000000000,0)--(4.00000000000000,0.000000000000000);
\draw[green!80,thin](5.50000000000000,0.000000000000000)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](6.00000000000000,-0.866000000000000)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(7.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,-1.73200000000000)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(6.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.50000000000000,0)--(8.50000000000000,-1.73200000000000);
\draw[green!80,thin](7.00000000000000,-2.59800000000000)--(8.00000000000000,-2.59800000000000);
\draw[green!80,thin](6.50000000000000,0)--(6.00000000000000,-0.866000000000000);
\draw[green!80,thin](8.50000000000000,0)--(9.00000000000000,-0.866000000000000);
\draw[green!80,thin](7.50000000000000,-3.46400000000000)--(7.50000000000000,-3.46400000000000);
\draw[green!80,thin](5.50000000000000,0)--(5.50000000000000,0.000000000000000);
\draw[green!80,thin](9.50000000000000,0)--(9.50000000000000,0.000000000000000);
\draw[green!80,thin](11.0000000000000,0.000000000000000)--(15.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.5000000000000,-0.866000000000000)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(12.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,-1.73200000000000)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(12.0000000000000,-1.73200000000000);
\draw[green!80,thin](13.0000000000000,0)--(14.0000000000000,-1.73200000000000);
\draw[green!80,thin](12.5000000000000,-2.59800000000000)--(13.5000000000000,-2.59800000000000);
\draw[green!80,thin](12.0000000000000,0)--(11.5000000000000,-0.866000000000000);
\draw[green!80,thin](14.0000000000000,0)--(14.5000000000000,-0.866000000000000);
\draw[green!80,thin](13.0000000000000,-3.46400000000000)--(13.0000000000000,-3.46400000000000);
\draw[green!80,thin](11.0000000000000,0)--(11.0000000000000,0.000000000000000);
\draw[green!80,thin](15.0000000000000,0)--(15.0000000000000,0.000000000000000);
\node at (0.000000000000000,-0.000000000000000) {$2$};
\node at (1.00000000000000,-0.000000000000000) {$4$};
\node at (2.00000000000000,-0.000000000000000) {$2$};
\node at (3.00000000000000,-0.000000000000000) {$4$};
\node at (4.00000000000000,-0.000000000000000) {$2$};
\node at (0.500000000000000,-0.866000000000000) {$2$};
\node at (1.50000000000000,-0.866000000000000) {$4$};
\node at (2.50000000000000,-0.866000000000000) {$2$};
\node at (3.50000000000000,-0.866000000000000) {$4$};
\node at (1.00000000000000,-1.73200000000000) {$2$};
\node at (2.00000000000000,-1.73200000000000) {$4$};
\node at (3.00000000000000,-1.73200000000000) {$2$};
\node at (1.50000000000000,-2.59800000000000) {$2$};
\node at (2.50000000000000,-2.59800000000000) {$4$};
\node at (2.00000000000000,-3.46400000000000) {$2$};
\node at (5.50000000000000,-0.000000000000000) {$1$};
\node at (6.50000000000000,-0.000000000000000) {$2$};
\node at (7.50000000000000,-0.000000000000000) {$4$};
\node at (8.50000000000000,-0.000000000000000) {$2$};
\node at (9.50000000000000,-0.000000000000000) {$4$};
\node at (6.00000000000000,-0.866000000000000) {$2$};
\node at (7.00000000000000,-0.866000000000000) {$1$};
\node at (8.00000000000000,-0.866000000000000) {$2$};
\node at (9.00000000000000,-0.866000000000000) {$4$};
\node at (6.50000000000000,-1.73200000000000) {$4$};
\node at (7.50000000000000,-1.73200000000000) {$2$};
\node at (8.50000000000000,-1.73200000000000) {$1$};
\node at (7.00000000000000,-2.59800000000000) {$2$};
\node at (8.00000000000000,-2.59800000000000) {$4$};
\node at (7.50000000000000,-3.46400000000000) {$4$};
\node at (11.0000000000000,-0.000000000000000) {$1$};
\node at (12.0000000000000,-0.000000000000000) {$1$};
\node at (13.0000000000000,-0.000000000000000) {$4$};
\node at (14.0000000000000,-0.000000000000000) {$1$};
\node at (15.0000000000000,-0.000000000000000) {$1$};
\node at (11.5000000000000,-0.866000000000000) {$1$};
\node at (12.5000000000000,-0.866000000000000) {$4$};
\node at (13.5000000000000,-0.866000000000000) {$1$};
\node at (14.5000000000000,-0.866000000000000) {$1$};
\node at (12.0000000000000,-1.73200000000000) {$4$};
\node at (13.0000000000000,-1.73200000000000) {$1$};
\node at (14.0000000000000,-1.73200000000000) {$1$};
\node at (12.5000000000000,-2.59800000000000) {$1$};
\node at (13.5000000000000,-2.59800000000000) {$1$};
\node at (13.0000000000000,-3.46400000000000) {$1$};
\draw[black,thin] (-0.5,0.5)--(15.5,0.5)--(15.5,-3.964)--(-0.5,-3.964)--(-0.5,0.5);
\end{tikzpicture}
%\caption{The interlacing triangular array $T \in \T_{4,5}$ (top) and the arrays $\splt(T)=(R,S) \in \T_{3,5} \times \T_{3,5}$ (bottom).}
%\label{fig:proof-merge-example}
%\end{figure}
\end{center}
\end{ex}

\subsection{Puzzle conversion}
\label{sec:puzzle-conversion}
A \emph{$0/1/10$-puzzle} is a labeling of the edges of $\Delta_n$ with labels $0$, $1$, and $10$ so that each $\Delta$-oriented face is labeled in one of the following ways

\begin{center}
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$1$}};
\end{tikzpicture} \quad
\begin{tikzpicture}[scale=1, rotate=-60]
    \node[circle, fill=black, scale=0.1] (c0) at ({-2*sqrt(3)/3},2) {};
    \node[circle, fill=black, scale=0.1] (d0) at ({-3*sqrt(3)/3},3) {};
    \node[circle, fill=black, scale=0.1] (d1) at ({-1*sqrt(3)/3},3) {};
    \draw[cyan!90,thin] (c0) -- (d0) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (c0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (d0) -- (d1) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture},
\end{center}
and so that each $\nabla$-oriented face is labeled by a $180^{\circ}$ rotation of one of these\footnote{See \cite[Section~5]{Knutson-Zinn-Justin-3} for the relationship between these puzzles and others which have appeared in the literature.}. We write $\tilde{\mc{P}}_n(\bs{\xi})$ for the set of $0/1/10$-puzzles on $\Delta_n$ with boundary conditions $\bs{\xi}$.

We are grateful to Allen Knutson for sharing with us the following correspondence between $1/2/3$-puzzles and $0/1/10$-puzzles. Given $\bs{\lambda}=(\lambda^{(1)},\lambda^{(2)},\lambda^{(3)})$, let $\partn(\bs{\lambda})$ be the $0,1$-strings $(\xi^{(1)},\xi^{(2)},\xi^{(3)})$ obtained by applying the transformation below:

%\begin{figure}
    \begin{center}
    \begin{tikzpicture}[scale=.75]
    \draw[blue!80,thin] (0,0) -- ({sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (1,0) -- ({1+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$3$}};
    \draw[blue!80,thin] (2,0) -- ({2+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (6.5,0.42) -- ({6.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$3$}};
    \draw[blue!80,thin] (8,.42) -- ({8+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (9.5,.42) -- ({9.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (3.5,1) -- ({3.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[blue!80,thin] (4.5,1) -- ({4.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$2$}};
    \draw[blue!80,thin] (5.5,1) -- ({5.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$3$}};
\end{tikzpicture}
\\ $\updownarrow$ \\
\begin{tikzpicture}[scale=.75]
    \draw[cyan!90,thin] (0,0) -- ({sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (1,0) -- ({1+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (2,0) -- ({2+sqrt(3)/3},1) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (6.5,0.42) -- ({6.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (8,.42) -- ({8+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (9.5,.42) -- ({9.5+2*sqrt(3)/3},.42) node [midway] {\large\textcolor{black}{$10$}};
    \draw[cyan!90,thin] (3.5,1) -- ({3.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$0$}};
    \draw[cyan!90,thin] (4.5,1) -- ({4.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$1$}};
    \draw[cyan!90,thin] (5.5,1) -- ({5.5+sqrt(3)/3},0) node [midway] {\large\textcolor{black}{$10$}};
\end{tikzpicture}.
\end{center}

\begin{prop}
\label{prop:puzzle-cryptomorphism}
For any boundary condition $\bs{\lambda}$, the transformation of edge labels shown above determines a bijection $\mc{P}_n(\bs{\lambda}) \to \tilde{\mc{P}}_n(\partn(\bs{\lambda}))$, where $\toprow$ denotes the inverse to $\partn$.
\end{prop}


\acknowledgements{We are grateful to Richard Stanley and Alexei Borodin for introducing us to this problem. We also thank Allen Knutson and Paul Zinn-Justin for their insightful comments.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}
