%% 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[final]{FPSAC2022}
%% 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[mathscr]{euscript} % make a nice \Net

\RequirePackage{cleveref}
%\usepackage{hypcap}
%\hypersetup{colorlinks=true, citecolor=darkblue, linkcolor=darkblue}
\definecolor{darkblue}{rgb}{0.0,0,0.7}
\newcommand{\darkblue}{\color{darkblue}}

\definecolor{darkred}{rgb}{0.68,0,0}
\newcommand{\darkred}{\color{darkred}}


\definecolor{darkgreen}{rgb}{0,.38,0}
\newcommand{\darkgreen}{\color{darkgreen}}

\newcommand{\defn}[1]{\emph{\darkblue #1}}
\newcommand{\defna}[1]{\emph{\darkred #1}}
\newcommand{\defnb}[1]{\emph{\darkblue #1}}
\newcommand{\defng}[1]{\emph{\darkgreen #1}}

\newcommand{\roma}[1]{{\darkred #1}}
\newcommand{\romb}[1]{{\darkblue #1}}
\newcommand{\romg}[1]{{\darkgreen #1}}

\newcommand{\defngb}[1]{{\textbf{\textit{\darkgreen #1}}}}

%%%%%%%%%%%%%%%%%% Theorem environment
\newtheorem{thm}{Theorem}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{claim}[thm]{Claim}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{conv}[thm]{Convention}
\newtheorem{question}[thm]{Question}
\newtheorem{por}[thm]{Porism}
\newtheorem{result}[thm]{Result}
\newtheorem{exc}{Exercise}[section]
\newtheorem{sublemma}[thm]{Sublemma}

% Theorem environments with roman or slanted font
\theoremstyle{remark}
\newtheorem{ex}[thm]{Example}
\newtheorem{nex}[thm]{Non-example}
\newtheorem{rem}[thm]{Remark}
%\newtheorem{rem}[thm]{Example}
\theoremstyle{definition}
\newtheorem{definition}[thm]{Definition}
%\newtheorem{Alg}[thm]{Algorithm}
%\newtheorem{Rtne}[thm]{Routine}
\newtheorem{Warn}[thm]{Caution}

\usepackage{lipsum}

%%%%%%%%%%%%% shortcuts and macros

\def\wh{\widehat}
\def\bu{\bullet}
\def\emp{\nothing}
\def\sq{\square}
%\def\st{\star}
%\def\bd{\blacktriangledown}
%\def\ap{\approx}
\def\RP{\mathbb R \mathbb P}
%\def\aA{\mathbb A}
\def\zz{\mathbb Z}
\def\AA{\mathbb A}
\def\nn{\mathbb N}
\def\cc{\mathbb C}
\def\ll{{L}}
\def\rr{\mathbb R}

\def\rrs{\mathbb R_{>0}}
\def\rrp{\mathbb R_{\ge 0}}

\def\qqq{\mathbb Q}
\def\tT{\mathbb T}
\def\ee{\mathbb E}
\def\hH{\mathbb H}
\def\ff{\mathbb F}
\def\sS{\mathbb S}
\def\SS{\mathbb S}
\def\oo{\mathbb O}
\def\pp{\mathbb P}
\def\fq{{\mathbb F}_q}
\def\ov{\overline}
\def\gG{\mathbb G}
\def\sm{\smallsetminus}
\def\pr{\prime}
\def\Ga{\Ups}
\def\Om{\Omega}
\def\De{\Delta}
\def\Si{\Sigma}
\def\la{\lambda}
\def\ga{\gamma}
\def\si{\sigma}
\def\de{\delta}
\def\ep{\epsilon}
\def\al{\alpha}
\def\be{\beta}
\def\om{\omega}
\def\k{\kappa}
%\def\ka{\vk}
\def\ve{\varepsilon}
\def\vr{\text{\rm deg}}
\def\vp{\varphi}
\def\vk{\varkappa}
%\def\CR{\mathcal R}
%\def\GG{\mathfrak G}
\def\GI{\mathfrak I}
\def\CS{\mathcal S}
\def\cC{\mathcal C}
\def\cB{\mathcal B}
\def\ca{\mathcal A}
\def\cA{\mathcal A}
%\def\cb{\mathcal B}
\def\cd{\mathcal D}
\def\ce{\mathcal E}
\def\cf{\mathcal F}
\def\cF{\mathcal F}
\def\cg{\mathcal G}
\def\cH{\mathcal H}
\def\cI{\mathcal I}
\def\ck{\mathcal K}
\def\cl{\mathcal L}
\def\cL{\mathcal L}
\def\cM{\mathcal{M}}


\def\co{\mathcal O}
\def\cp{\mathcal P}
\def\cP{\mathcal P}
\def\cQ{\mathcal Q}
\def\cR{\mathcal R}
\def\cT{\mathcal T}
\def\cv{\mathcal V}
\def\cw{\mathcal W}
\def\CB{\mathcal B}
\def\T{\mathbf{T}}
\def\CT{\mathcal T}
\def\BB{\mathbf{B}}
\def\bP{\mathbf{P}}
\def\ssu{\subset}
\def\sss{\supset}
%\def\wt{\widetilde}
\def\<{\langle}
\def\>{\rangle}

\def\DET{\text{{\rm DET}}}

\def\const{\text{{\rm const}}}
\def\width{\text{{\rm width}}}
%\def\sc{ {\text {\rm sc}  } }
\def\rc{ {\text {\rm rc}  } }
\def\y{ {\text {\rm y}  } }
\def\sy{ {\text {\rm sy}  } }
%\def\Z{ {\text {\rm Z} } }
\def\Z{\zz}
\def\FF{ {\text {\rm F} } }
\def\RR{ {\text {\rm R} } }
\def\SO{ {\text {\rm SO} } }
\def\OOO{ {\text {\rm O} } }
\def\SL{ {\text {\rm SL} } }
\def\GL{ {\text {\rm GL} } }
\def\NF{ {\text {\rm NF} } }
\def\rR{ {\text {\rm R} } }
\def\rM{ {\text {\rm M} } }

\def\rI{ {\text {\sc I} } }
\def\rJ{ {\text {\sc J} } }
\def\ri{ {\text {\rm i} } }
%\def\rH{ {\text {\sc H} } }

\def\rA{ {{A} } }
\def\rC{ {{C} } }
\def\rP{ {\text {\rm P} } }
\def\Ker{ {\text {\rm Ker} } }
\def\Cat{ {\text {\rm Cat} } }

\def\height{ {\text {\rm height} } }

\def\rk{\textnormal{rk}}
\def\girth{\textnormal{girth}}
\def\polygirth{\textnormal{polygirth}}

\def\srk{\dim}
\def\prev{\prec}
\def\oa{\overrightarrow}
\def\Ups{\Upsilon}
\def\ups{\upsilon}
\def\gar{r}
\def\vt{\ze}
\def\dv{\divideontimes}
\def\U{{\text {\rm U} } }
\def\rT{{\text {\rm T} } }
\def\Q{{\text {\rm Q} } }
\def\mix{{\text {\rm mix} } }
\def\Sf{{\text{{\it \bf S}}}}
\def\rK{{\text{\textsf{K}}}}
\def\RU{\UR}

\def\St{{\text {\rm Stab} } }
\def\rkm{{f}}
\def\rkn{{g}}
\def\Fn{{\Phi}}
\def\Mor{{\Phi}}



\def\0{{\mathbf 0}}
\def\id{{\rm id}}
\def\st{{\rm st}}
\def\th{{\rm th}}
\def\ol#1{{\overline {#1}}}
\def\tl{\triangleleft}
%\def\ED{{\text{\rm ED}}}
\def\II{{\mathcal I}}

%\def\par{{\partial}}
\def\LL{{\mathbb{L}}}
\def\NN{{\mathbb N}}
\def\OO{{\mathcal O}}
%\def\TT{{\mathcal T}}
\def\VV{{\mathcal V}}
\def\YY{{\mathcal Y}}

\def\closest{\multimap}
\def\implies{\Rightarrow}
\def\nothing{\varnothing}

\def\.{\hskip.06cm}
\def\ts{\hskip.03cm}
\def\tts{\hskip.01cm}
\def\sts{\hskip.015cm}
\def\mts{\hspace-.04cm}
\def\ang{\angle \.}
\def\sphang{\sphericalangle}
\def\sang{\sphericalangle \.}

\def\dist{{\text {\rm dist} }}
\def\area{{\text {\rm area} }}
\def\per{{\text {\rm perimeter} }}
\def\vol{{\text {\rm vol}}}
\def\conv{{\text {\rm {conv}} }}
\def\shul{{\text {\rm {sh}} }}
\def\clip{{\text {\rm {clip}} }}
\def\radius{{\text {\rm radius}}}
\def\cent{{{\text{\rm cm}}}}
\def\diam{{\text {\rm diam}}}
\def\ind{{\text {\rm ind}}}
\def\cayley{{\text {\rm Cayley} }}
\def\CM{{\text {\rm CM}}}

\def\lra{\leftrightarrow}
\def\Lra{\Leftrightarrow}
\def\bow{\bowtie}
\def\zyl{\asymp}
\def\pt{\partial}
\def\nl{\textbf{\textit{u}}}
\def\nlH{\nl_{\hskip-.04cm H}}
\def\nlF{\nl_{\hskip-.04cm F}}
\def\bio{{\text{\bf {0}}}}
\def\bz{{\textbf{z}}}
\def\bx{{\textbf{x}}}
\def\by{{\textbf{y}}}
\def\nts{\hskip-.04cm}
%
\def\bbu{\textbf{\textit{u}}}
\def\bbe{\textbf{\textit{e}}}
\def\bbc{\textbf{\textit{c}}}
\def\bw{\textbf{\textit{w}}}
\def\bbw{\textbf{\textit{w}}}
\def\bbv{\textbf{\textit{v}}}
\def\br{\textbf{\textit{r}}}
%\def\vb{\textbf{\textit{a}}}
\def\bv{\textbf{\textit{v}}}

\def\ba{\textbf{\textit{a}}}
\def\bb{\textbf{\textit{b}}}
\def\bd{\textbf{\textit{d}}}
%\def\bbe{\textbf{\textit{e}}}

%\def\bA{\textbf{\textit{A}}}

\def\bX{\textbf{\textit{X}}}
\def\bZ{\textbf{\textit{Z}}}
\def\bcx{\textbf{\textit{x}}}
\def\bcy{\textbf{\textit{y}}}

%
\def\gR{\frak{A}}
\def\gD{\frak{D}}
\def\bal{{\mathbf{\al}}}
\def\vvt{f}
\def\bdel{\mathbf{\de}}
\def\ud{}
\def\mi{{\rm {\textsf{m}}}}

\def\cyR{{$(\Re)$}}
\def\mcc{{($\frak{J})$}}
\def\corn{{($\frak{C})$}}
\def\mich{{($\frak{M})$}}

\def\La{\Lambda}
\def\LA{\Lambda}
\def\LAM{\Lambda^\triangledown}

\def\btr{{\rho}}
\def\ctr{{\mathfrak{R}}}
\def\Ctr{{\mathcal{A}}}
%\def\bbu{\text{\bf{{e}}}}
%\def\bw{\text{\bf{{w}}}}
%\def\bw{\textbf{w}}
%\def\br{\textbf{r}}

\def\bba{{\text{\bf a}}}
\def\bbb{{\text{\bf b}}}
%\def\bbc{{\text{\bf c}}}

\def\di{{\small{\ts\diamond\ts}}}
\def\dQ{{\mathcal{Q}}}
\def\Di{{\mathfrak{D}}}
\def\kK{{\Bbbk}}
\def\ze{{\zeta}}

\def\senkin{{{Sen{\rm \small \textsf{'}}kin}}}
\def\fsenkin{{{Sen{\rm \footnotesize\textsf{'}}kin}}}

\def\dimo{{\diamond}}
\def\bstar{{\sharp}}

\def\open{{\rm \footnotesize [{$\ast$}] \.\ts\ts}}

\def\ram{{\!\! {\footnotesize $\lozenge$ \,}}}
\def\ra{{\rm {\footnotesize [\texttt{1}]} \.\ts }}
\def\rapu{{\rm {\footnotesize [\texttt{1+}]} \.\ts }}
\def\rami{{\rm {\footnotesize [\texttt{1-}]} \.\ts }}
%
\def\rapmu{{\rm {\footnotesize [\texttt{1-}]}, {\footnotesize [\texttt{1}]} \. and {\footnotesize [\texttt{1+}]}}}
\def\rbpmu{{\rm {\footnotesize [\texttt{2-}]}, {\footnotesize [\texttt{2}]} \. and {\footnotesize [\texttt{2+}]}}}
%
\def\rbmi{{\rm {\footnotesize [\texttt{2-}]} \.\ts }}
\def\rb{{\rm {\footnotesize [\texttt{2}]} \.\ts }}
\def\rbpu{{\rm {\footnotesize [\texttt{2+}]} \.\ts }}
% \def\rbpmu{{\rm {\footnotesize [\texttt{2$\pm$}]} \.\ts }}
% \def\rcmi{{\rm {\footnotesize [\texttt{3-}]} \.\ts }}
% \def\rd{{\rm {\footnotesize [\texttt{3}]} \.\ts }}
\def\rcmi{\rbpu}
\def\rd{\rbpu}

% \newcommand{\id}{\mathrm{id}}


\newcommand{\tr}{\mathrm{tr}}
\newcommand{\wi}{\mathrm{width}}
\newcommand{\EA}{\mathcal{A}}
\newcommand{\BT}{\mathcal{BT}}
\newcommand{\PT}{\mathcal{PT}}
%\newcommand{\LR}{\mathcal{LR}}
\newcommand{\PP}{\operatorname{PP}}
\newcommand{\Hex}{\operatorname{H}}
\newcommand{\RPP}{\operatorname{RPP}}
\newcommand{\SSYT}{\operatorname{SSYT}}
\newcommand{\SYT}{\operatorname{{\rm SYT}}}



\def\nin{\noindent}
\def\tts{\hskip.02cm}


\def\SS{{S}}
\def\bal{\ov{\al}}
%\def\bbe{\ov{\be}}
\def\hba{\hslash}

\def\qp{\textrm{q}}
\def\eG{\Phi}
\def\SP{{\textsc{\#P}}}
\def\NP{{\textsc{NP}}}
\def\SP{{\textsc{\#{}P}}}
\def\GapP{{\textsc{GapP}}}
\def\poly{{\textsc{P}}}

%\def\Pb{\pp}
\def\Pb{{\text{\bf P}}}
\def\Path{{\text{\rm Path}}}

\def\aD{\textrm{D}}
\def\aDr{\textrm{\em D}}
\def\aE{\textrm{E}}
\def\aEr{\textrm{\em E}}
\def\aF{\textrm{F}}
\def\aFr{\textrm{\em F}}

\def\aG{{\beth}}
\def\aGr{{\aG}}

\def\aH{\textrm{H}}
\def\aHr{\textrm{\em H}}
\def\ah{\textrm{h}}
\def\aI{{ \textrm{I} } }
\def\aIr{\textrm{\em I}}
\def\aJ{\textrm{J}}
\def\aJr{\textrm{\em J}}
\def\aK{\textrm{K}}
\def\aKr{\textrm{\em K}}
\def\aL{\textrm{L}}
\def\aLr{\textrm{\em L}}
\def\aM{\textrm{M}}
\def\am{\textrm{m}}
\def\aMr{\textrm{\em M}}
\def\aN{\textrm{N}}
\def\aNr{\textrm{\em N}}
\def\an{\textrm{n}}

\def\sn{{{\textsc{n}}}}


\def\aP{\textrm{p}}
\def\aPr{\textrm{\em p}}

\def\aR{\textrm{R}}
\def\aRr{\textrm{\em R}}
\def\ar{d}
\def\aU{\textrm{U}}
\def\aUr{\textrm{\em U}}
\def\av{\textrm{v}}


\def\bA{\textbf{\textrm{A}}\hskip-0.03cm{}}
\def\bAr{\textbf{\textrm{\em A}}\hskip-0.03cm{}}

\def\bB{\textbf{\textrm{B}}\hskip-0.03cm{}}
\def\bBr{\textbf{\textrm{\em B}}\hskip-0.03cm{}}

\def\bC{\textbf{\textrm{C}}\hskip-0.03cm{}}
\def\bD{\textbf{\textrm{D}}\hskip-0.03cm{}}
\def\bE{\textbf{\textit{E}}\hskip-0.03cm{}}
\def\bF{\textbf{\textit{F}}\hskip-0.03cm{}}
\def\bI{\textbf{\textit{I}}\hskip-0.03cm{}}
\def\bL{\textbf{\textit{L}}\hskip-0.03cm{}}
\def\bM{\textbf{\textrm{M}}\hskip-0.03cm{}}

\def\bMr{\textbf{\textrm{\em M}}\hskip-0.03cm{}}
\def\bNr{\textbf{\textrm{\em N}}\hskip-0.03cm{}}

\def\bN{\textbf{\textrm{N}}\hskip-0.03cm{}}
\def\bS{\textbf{\textit{S}}{}}
%\def\bT{\textbf{\textit{T}}\hskip-0.03cm{}}
\def\bW{\textbf{\textit{W}}\hskip-0.03cm{}}

\def\bG{\textbf{\textit{G}}\hskip-0.03cm{}}
\def\bH{\textbf{\textit{H}}\hskip-0.03cm{}}
\def\bJ{\textbf{\textit{J}}\hskip-0.03cm{}}
\def\bK{\textbf{\textrm{K}}\hskip-0.03cm{}}


%\def\bT{\text{\bf T}}
\def\bT{\mathbf{T}}



\def\ana{\emph{\textsf{a}}}
\def\bnb{\emph{\textsf{b}}}
\def\cnc{\emph{\textsf{c}}}
\def\dnd{\emph{\textsf{d}}}
\def\bnbt{{\textsf{b}}}
\def\dndt{{\textsf{d}}}

%\def\qnq{\emph{\textsf{q}}}
\def\qnq{q}
\def\rnr{t}


%\def\rnr{\mathfrak{r}}
\def\rnrt{\textsf{r}}
\def\unu{\emph{\textsf{u}} \ts}
\def\vnv{\emph{\textsf{v}} \ts}
\def\wnw{\emph{\textsf{w}} \ts}
\def\wnwr{\emph{\textsf{w}}}

\def\A{A}
\def\B{B}
\def\C{C}
\def\D{D}
\def\E{E}

\def\aAr{\mathrm{A}}
\def\aBr{\mathrm{B}}
\def\aCr{\mathrm{C}}
\def\aA{\textrm A}
\def\aB{\textrm B}
\def\aC{\textrm C}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Macros by Swee Hong


\DeclareMathOperator{\Abar}{\overline{\bA}}
\DeclareMathOperator{\Ac}{\mathcal{A}}
\DeclareMathOperator{\Bc}{\mathcal{B}}
\def\aa{\textrm{a}}
\def\ab{\textrm{b}}
\def\abr{\text{\em b}}



% \def\ac{\textrm{c}}
% \def\acr{\mathrm{c}}

\def\ac{c}
\def\acr{c}
\DeclareMathOperator{\Act}{\textnormal{Act}}

\def\ad{\textrm{d}}
\def\ag{\textrm{g}}
\def\af{\textrm{f}}

%\def\ap{\textrm{p}}
%\def\apr{\textrm{\em p}}

\def\ap{\om}
\def\apr{\om}


\def\aq{\textrm{q}}
\def\aqr{\textrm{\em q}}
\def\arr{\textrm{\em r}}
\def\as{\textrm{s}}
\def\asr{\textrm{\em s}}
\def\au{\textrm{u}}
\def\aw{\textrm{w}}

\def\Low{\textrm{\em Low}}


\newcommand\agw[1]{\widehat{\mathrm{g}}^{\<#1\>}}
\newcommand\ahw[1]{\widehat{\mathrm{h}}^{\<#1\>}}
\newcommand\avw[1]{\widehat{\mathrm{v}}^{\<#1\>}}
\newcommand\aww[1]{\widehat{\mathrm{w}}^{\<#1\>}}



\DeclareMathOperator{\Aunt}{\textnormal{Aunt}}

\DeclareMathOperator{\Cc}{\mathcal{C}}
\DeclareMathOperator{\Child}{\textnormal{Child}}
\DeclareMathOperator{\cMb}{\overline{\cM}}

\newcommand\Class[1]{\textnormal{Cnt}_{#1}/\sim_{#1}}

\DeclareMathOperator{\Cov}{\textnormal{Cov}}
\DeclareMathOperator{\Cnt}{\textnormal{Cont}} %Set of feasible continuations
\DeclareMathOperator{\Dc}{\mathcal{D}}
\DeclareMathOperator{\depth}{\textnormal{depth}} %Set of feasible continuations

\DeclareMathOperator{\Des}{\textnormal{Des}}

\DeclareMathOperator{\eb}{\mathbf{e}}
\DeclareMathOperator{\Ec}{\mathcal{E}} %The set of all linear extensions
\DeclareMathOperator{\El}{\overline{\mathcal{E}}} %The set of all partial linear extensions
\DeclareMathOperator{\Fam}{\textnormal{Fam}}
\DeclareMathOperator{\Fb}{\mathbb{F}}
\DeclareMathOperator{\Fc}{\mathcal{F}}
\DeclareMathOperator{\Fin}{\textnormal{Comp}} %Set of finalizer
\DeclareMathOperator{\Hc}{\mathcal{H}}
\DeclareMathOperator{\Hr}{\mathscr{H}}
\newcommand{\Hf}{{\mathsf{H}}} %subspace
\newcommand{\hgt}{\textnormal{ht}}
\DeclareMathOperator{\Ic}{\mathcal{I}}
\DeclareMathOperator{\Jc}{\mathcal{J}}
\DeclareMathOperator{\Jcm}{{\mathcal{J}}\hskip-.0572cm}
\DeclareMathOperator{\kb}{\mathbf{k}}
\DeclareMathOperator{\Lc}{\mathcal{L}}
\DeclareMathOperator{\Uc}{\mathcal{U}}

\DeclareMathOperator{\Mbar}{\overline{\bM}}
\DeclareMathOperator{\Nbar}{\overline{\bN}}
% \DeclareMathOperator{\mb}{\mathbf{m}}

\DeclareMathOperator{\mb}{{\textbf{m}}}
\DeclareMathOperator{\nb}{\mathbf{n}}
\DeclareMathOperator{\Nb}{\mathbb{N}}
\DeclareMathOperator{\Nc}{\mathcal{N}}
\def\oa{\overline}
\DeclareMathOperator{\Oc}{\mathcal{O}} % Set of order-preserving functions
\DeclareMathOperator{\op}{\textnormal{op}} % Set of order-preserving functions
\DeclareMathOperator{\Par}{\textnormal{Par}}
\DeclareMathOperator{\Pas}{\textnormal{Pas}}
\newcommand{\precdot}{\prec\mathrel{\mkern-5mu}\mathrel{\cdot}}
\DeclareMathOperator{\Rb}{\mathbb{R}}
\newcommand{\salp}{\sim_\alpha} % equivalence relation
\newcommand{\satu}{\mathbf{1}} % vector of all 1
\newcommand{\Sc}{\mathcal{S}}
\newcommand{\spstar}{\textsf{null}} %special element
\newcommand{\spstarr}{{\textsf{\em null}}} %special element
\newcommand{\supp}{\textnormal{supp}}
\newcommand{\Stn}{\textnormal{Stn}} %Steiner system
\newcommand{\Uchild}{\textnormal{Uchild}}
\newcommand{\us}{u} % poset macro
\newcommand{\ws}{w} % poset macro

\DeclareMathOperator{\fb}{\mathbf{f}}
\DeclareMathOperator{\gb}{\mathbf{g}}
\newcommand\gbw[1]{\mathbf{\widehat{g}}^{\<#1\>}}
\DeclareMathOperator{\hb}{\mathbf{h}}

%\newcommand\hbw[1]{\mathbf{\widehat{h}}^{\<#1\>}}





\newcommand\hbw[1]{{\bT^{\<#1\>}\mathbf{{h}}}}


\DeclareMathOperator{\pb}{\mathbf{p}}
\DeclareMathOperator{\prt}{\textnormal{prt}}
\DeclareMathOperator{\rad}{\textnormal{rad}}

\DeclareMathOperator{\qb}{\mathbf{q}}
\DeclareMathOperator{\tb}{\mathbf{t}}
\DeclareMathOperator{\ub}{\mathbf{u}}
\DeclareMathOperator{\vb}{\mathbf{v}}
%\newcommand\vbw[1]{\mathbf{\widehat{v}}^{\<#1\>}}
%\newcommand\vbw[1]{\mathbf{{v}}^{\<#1\>}}

\newcommand\vbw[1]{\bT^{\<#1\>}\mathbf{{v}}}

\DeclareMathOperator{\wb}{\mathbf{w}}
\newcommand\wbw[1]{\mathbf{\widehat{w}}^{\<#1\>}}
\DeclareMathOperator{\wt}{\mathtt{wt}} % weight function
\DeclareMathOperator{\xb}{\mathbf{x}}
\def\Xf{{\wh{X}}}

\DeclareMathOperator{\yb}{\mathbf{y}}
\DeclareMathOperator{\zb}{\mathbf{z}}
\DeclareMathOperator{\Zb}{\mathbb{Z}}



\def\precc{\preccurlyeq} % nicer poset leq sign
\def\succc{\succcurlyeq} % nicer poset geq sign


\def\eone{\textbf{\ts\textrm{e}$_1$}}
\def\etwo{\textbf{\ts\textrm{e}$_2$}}


%%%%%%%%%%%%%%%%%%%%%%%%%    Macros for atlases

\DeclareMathOperator{\Ef}{\Theta} %  Edge set
\DeclareMathOperator{\ef}{\text{\it e}} %  edges
\DeclareMathOperator{\Qf}{ {\Gamma} } %  Quiver
\DeclareMathOperator{\Rf}{\mathscr{R}} %  Quiver
\DeclareMathOperator{\Vf}{\Omega} %  Vertex set
\DeclareMathOperator{\Vfm}{\Omega^0} %  Vertex set
\DeclareMathOperator{\Vfp}{\Omega^+} %  Vertex set

\DeclareMathOperator{\Wf}{\mathscr{W}} %  Vertex set
\DeclareMathOperator{\vf}{{\text{\it \sts v}}} %  vertex
\DeclareMathOperator{\vfs}{{\text{\it \sts v}}^\ast} %  vertex
%\DeclareMathOperator{\wf}{\mathfrak{w}} %  vertex
\DeclareMathOperator{\wf}{{\text{\it \sts w}}} %  vertex

\DeclareMathOperator{\inc}{\textnormal{inc}} % incomparable index

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     Macros for greedoid

\def\zero{O}
\def\cD{\Df}



\newcommand{\Df}{\mathscr{D}}
\newcommand{\Ff}{\mathscr{F}}
\newcommand{\Gf}{\mathscr{G}}
\newcommand{\Mf}{\mathscr{M}}
\newcommand{\Nf}{\mathscr{N}}

%%%%%%%%%%%%%%%%%%%%Macros for posets

\newcommand{\lcov}{{\leftarrowtail}}
\newcommand{\rcov}{{\rightarrowtail}}

%\newcommand{\Cov}{{\leftarrowtail}}


%\newcommand{\Pf}{\mathscr{P}}
\newcommand{\Pf}{\cP}
\newcommand{\Af}{\mathscr{A}}

\newcommand{\low}{\textnormal{down}}
\newcommand{\up}{\textnormal{up}}
\newcommand{\zf}{z}

\newcommand{\Zf}{{Z}}
\newcommand{\Zlow}{{Z}_{\textnormal{down}}}
\newcommand{\Zup}{{Z}_{\textnormal{up}}}

%\newcommand{\Zlow}{\mathcal{Z}_{\textnormal{down}}} % set of minimal elements
%\newcommand{\Zup}{\mathcal{Z}_{\textnormal{up}}} % set of minimal elements
%\newcommand{\Zf}{\mathcal{Z}} % set of minimal elements

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






%% define your title in the usual way
\title{Log-Concave Poset Inequalities: Extended Abstract}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[S.~H.~Chan and I.~Pak]{Swee Hong Chan\addressmark{1}\thanks{\href{mailto:sweehong@math.ucla.edu}{sweehong@math.ucla.edu}.  Partially supported by the Simons Foundation.} \and Igor Pak\addressmark{1}\thanks{\href{mailto:pak@math.ucla.edu}{pak@math.ucla.edu}.  Partially supported by the National Science Foundation.}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, UCLA, Los Angeles, CA}

%% 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 study combinatorial inequalities for various classes of set systems:
	matroids, morphisms of matroids, polymatroids, and poset antimatroids.
	We prove log-concave inequalities for counting certain weighted
	feasible words, which generalize and extend several previous results
	establishing \emph{Mason conjectures} for the numbers of independent sets
	of matroids.
	Additionally, we rederive \emph{Stanley's
		inequality} on the number of certain linear extensions, which we then also extend to the weighted case.
	  Notably, we also prove matching equality
	conditions for all these inequalities.
}

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%\keywords{Log-concavity, matroids, greedoids, posets, linear extensions}

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\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



%\subsection{Foreword}
%It is always remarkable and even a little suspicious, when a nontrivial
%property can be proved for a large class of objects.  Indeed, this says
%that the result is ``global'', i.e.\ the property is a consequence of the
%underlying structure rather than individual objects.  Such results are
%even more remarkable in combinatorics, where the structures are weak
%and the objects are plentiful.  In fact, many reasonable conjectures
%in the area fail under experiments, while some are ruled out by
%theoretical considerations.


%\bigskip

\nin
{\bf Foreword:} This extended abstract is concerned with log-concavity results for counting problems
in the general context of posets, and is motivated by a large body of
amazing recent work in area, see a survey by Huh~\cite{Huh} and also earlier surveys
\cite{Bra,Bre,Bre2,Sta2}.
We select a subset of these results for various combinatorial structures
to be presented here.
Their proofs and historical remarks can  be found in the full version of the paper~\cite{CP}, and a simpler version of the method can be found in the expository version of the paper~\cite{CP2}.

\section{Matroids}

A (finite) \defna{matroid}~$\Mf$ is a pair $(X,\Ic)$ of a
\defnb{ground set} $X$, $|X|=n$, and a nonempty collection of \defnb{independent sets}
$\Ic \subseteq 2^X$ that satisfies the following:
\begin{itemize}
	%	\item \, $\emp\in \Ic$,
	\item (\defng{hereditary property}) \, $S\subset T$, \. $T\in \Ic$ implies $S\in \Ic$, and
	\item (\defng{exchange property}) \,  $S, T\in \Ic$, \.  $|S|<|T|$ implies
	there exists $ x \in T \setminus S$ such that $S+x \in \Ic$.
\end{itemize}
\defnb{Rank} of a matroid is the maximal size of the independent set: $\rk(\Mf) := \max_{S\in \Ic} |S|$.
%
A \defnb{basis} of a matroid is an independent set of size $\rk(\Mf)$.  Finally,
let $\cI_k := \bigl\{S\in \cI, \. |S|=k\bigr\}$,
and let $\rI(k)=\bigl|\cI_k\bigr|$ be the \defn{number of independent sets}
in $\Mf$ of size~$k$, $0\le k \le \rk(\Mf)$.


%\subsection{Mason's conjecture}
%\smallskip

Mason's matroid log-concavity conjectures were stated in~\cite{Mas},
motivated by the earlier work and conjectures in graph theory and
combinatorial geometry.
The strongest form of these conjectures were recently proved independently
by Br\"and\'en and Huh~\cite{BH} through \emph{Lorentzian polynomials} approach, and by Anari~\textit{et al.}~\cite{ALOV}
through \emph{completely log-concave polynomials} approach.

% log-concavity
%of the \defng{characteristic polynomial} of a matroid, which is a generalization
%of the graph chromatic polynomial, and a specialization of the Tutte polynomial.
%They then deduce the Welsh--Mason Conjecture~\eqref{eq:matroid-LC} indirectly, via
%an observation by Brylawski~\cite{Bry1} (see also~\cite{Lenz2}).

%%\smallskip
%
%\begin{thm}[{\rm \defng{Log-concavity for matroids}, \cite[Theorem 9.9~(3)]{AHK},
%		formerly \defna{Welsh--Mason conjecture}}{}] \label{t:matroids-AHK}
%	For a matroid  $\Mf=(X,\Ic)$  and integer  $1\le k < \rk(\Mf)$, we have:
%	\begin{equation}\label{eq:matroid-LC}
%		\rI(k)^2 \. \ge \. \rI(k-1)  \cdot  \rI(k+1).
%	\end{equation}
%\end{thm}
%
%%\smallskip
%
%\nin Further improvements to~\eqref{eq:matroid-LC} have been long conjectured
%by Mason~\cite{Mas} and were recently established in quick succession.
%
%%\smallskip
%
%\begin{thm}[{\rm  \defng{One-sided ultra-log-concavity for matroids}, \cite[Corollary~9]{HSW}, formerly \defna{weak Mason conjecture}}{}]
%	\label{t:matroids-HSW}
%	For a matroid  $\Mf=(X,\Ic)$  and integer  $1\le k < \rk(\Mf)$, we have:
%	\begin{equation}\label{eq:matroid-ULC-one-sided}
%		\rI(k)^2 \, \ge \, \left(1 \. + \, \frac{1}{k}\right)   \,  \rI(k-1)  \   \rI(k+1).
%	\end{equation}
%\end{thm}
%
%
%%\smallskip

\begin{thm}[{\defng{Ultra-log-concavity for matroids}~\cite[Theorem~1.2]{ALOV} and~\cite[Theorem~4.14]{BH}, formerly \defna{strong Mason conjecture}{}}]
	\label{t:matroids-BH}
	For a matroid  $\Mf=(X,\Ic)$,   $|X|=n$,   and integer  $1\le k < \rk(\Mf)$, 
	\begin{equation}\label{eq:matroid-ULC}
		\rI(k)^2 \, \ge \, \left(1 \. + \, \frac{1}{k}\right)  \left(1 \. + \, \frac{1}{n-k}\right)  \,\.  \rI(k-1)  \   \rI(k+1).
	\end{equation}
\end{thm}

%\smallskip

%\nin
Mason's conjectures were presented as a series of three log-concave inequalities,
for which \eqref{eq:matroid-ULC} is the strongest version of the inequalities.
The other two inequalities were previously obtained in \cite{AHK} and \cite{HSW}
 by using   the hard Lefschetz theorem and the Hodge--Riemann
relations in a number of algebraic settings.
   The methods in \cite{ALOV,BH} use interrelated ideas, and
avoid much of the algebraic technology in~\cite{AHK}.


Equality conditions for \eqref{eq:matroid-ULC}
were recently established by Murai, Nagaoka and Ya\-za\-wa
in~\cite{MNY} using an algebraic argument built on~\cite{BH}, and can be described as follows.
For a matroid  $\Mf=(X,\Ic)$  on  $|X|=n$  elements,
define \. $\girth(\Mf)  :=  \min\bigl\{k\.:\. \rI(k)< \binom{n}{k}\bigr\}$.
The  \defn{girth of a matroid}
is the size of the smallest non-independent sets in~$\Mf$.

%\smallskip

\begin{thm}[{\defng{Equality for matroids}~\cite[Corollary~1.2]{MNY}}{}] \label{t:matroids-equality}
%	Let  $\Mf=(X,\Ic)$  be a matroid on  $|X|=n$ elements,
%	and let  $1\le k < \rk(\Mf)$.
	Equality  occurs in \eqref{eq:matroid-ULC}
%	\begin{equation}\label{eq:matroid-ULC-equality}
%		\rI(k)^2 \, = \, \left(1\. + \. \frac{1}{k}\right)  \left(1\. + \, \frac{1}{n-k}\right)  \,\.  \rI(k-1)  \   \rI(k+1)
%	\end{equation}
	\underline{if and only if} we have $\girth(\Mf)>(k+1)$.
\end{thm}

%\smallskip



%\com{SH}{Note for self, add historical stuffs, if there are still spaces left.}

%\subsection{Refined log-concavity for matroids}\label{ss:intro-matroids}

We now present a refinement of log-concavity for matroids that is proved through our method.
For an independent set  $S\in \cI$  of a matroid  $\Mf=(X,\Ic)$, denote by
%
\begin{equation}\label{eq:matroid-cont-def}
	\Cnt(S) \, := \bigl\{x \in X \setminus S~:~S +x \in \cI\bigr\}
\end{equation}
%
the set of \defnb{continuations} of~$S$. For all  $x, y\in \Cnt(S)$, we write 
$x\sim_S y$  when  $S+x+y \notin\cI$. Note that  ``$\sim_S$'' is an equivalence
relation.
We call an equivalence class of the relation  $\sim_S$  a \defn{parallel class} of~$S$,
and we denote by $\Par(S)$ the set of parallel classes of~$S$.


For every  $0\le k< \rk(\Mf)$, define the  \defnb{$k$-continuation number} of a matroid~$\Mf$
as the maximal  number of parallel classes of independent sets of size~$k$:
%
\begin{equation}\label{eq:matroid-p-def}
	\aP(k) \ := \ \max   \bigl\{ \.  \big|\Par(S)\big|  \ : \  S \in \Ic_k \. \bigr\}.
\end{equation}
%
Clearly,  $\aP(k)\le n-k$.

%\smallskip

\begin{thm} [{\defng{Refined log-concavity for matroids}}{}]
	\label{t:matroids-Par}
	For a matroid  $\Mf=(X,\Ic)$  and integer  $1\le k < \rk(\Mf)$, we have:
	\begin{equation}\label{eq:matroid-Par}
		\rI(k)^2 \, \ge \, \left(1\. + \. \frac{1}{k}\right)  \left(1\. + \, \frac{1}{\aPr(k-1)-1}\right)  \,\.  \rI(k-1)  \   \rI(k+1).
	\end{equation}
\end{thm}

%\smallskip

%\nin
Clearly, Theorem~\ref{t:matroids-Par} implies Theorem~\ref{t:matroids-BH}.
In particular,  let us illustrate the power of this refinement in a special case.


	Let  $G=(V,E)$  be a connected graph
with  $|V|=\sn$  edges.  The corresponding \defnb{graphical matroid}  $\Mf_G=(E,\cI)$
 is defined to have independent sets to be all \emph{forests} in~$G$.  Then  $\rI(k)$  is the number of 
forests with $k$ edges.

%\smallskip

\begin{prop}[{\defng{Refinement for graphical matroids}}{}]\label{p:graphical}
	Let  $G=(V,E)$  be a simple connected graph on  $|V|=\sn$  vertices, and let
	 $\aIr(k)$  be the number of  forests with $k$ edges.  Then
	\begin{equation}\label{eq:prop-graphical-equality}
		\frac{\aIr(\sn-2)^2}{\aIr(\sn-3) \. \cdot \. \aIr(\sn-1)}  \  \ge \  \frac{3}{2} \. \bigg(1\. + \. \frac{1}{\sn-2} \bigg).
	\end{equation}
%	and the equality holds \, \underline{if and only if} \, $G$  is an  $\sn$-cycle.
	% Furthermore, for all  $1\le k \le \sn-3$,
	% the refined log-concavity inequality~\eqref{eq:matroid-Par} for  $\Mf_G$  is always strict.
\end{prop}

%\smallskip

%\nin
This is both numerically and asymptotically better than~\eqref{eq:matroid-ULC}.
For example, when \. $|E|-\sn \to \infty$, we have:
\[
\frac{\aI(\sn-2)^2}{\aI(\sn-3) \. \cdot \. \aI(n-1)}  \  \geq_{\eqref{eq:matroid-ULC}} \  \bigg(1 \. + \. \frac{1}{|E|-\sn +2} \bigg) \.
\bigg(1 \. +\. \frac{1}{\sn-2} \bigg) \ \to \ 1 \quad \text{as} \quad \sn\to\infty.
\]

%\smallskip

We now  present equality conditions for the refined log-concave inequality~\eqref{eq:matroid-Par}.

\begin{thm}[{\defng{Refined equality for matroids}}{}]
	\label{t:matroids-equality-Par}
	%	Let  $\Mf=(X,\Ic)$ be a matroid, and let  $1\le k < \rk(\Mf)$. Then:
	%	\begin{equation}\label{eq:matroid-equality-Par}
	%		\rI(k)^2 \, = \, \left(1\. + \. \frac{1}{k}\right)  \left(1\. + \, \frac{1}{\aPr(k-1)-1}\right)  \,\.  \rI(k-1)  \   \rI(k+1)
	%	\end{equation}
	Equality occurs in \eqref{t:matroids-Par} \,
	\underline{if and only if} \,
	there exists \. $\asr(k-1)>0$ such that  for every  $S\in \cI_{k-1}$  we have:
	\begin{align}
		\big|\Par(S)\big| \ &= \  \aPr(k-1)\.,  \quad \ \text{and} \label{eq:ME1}
		\tag{ME1}\\
		|\Cc|  \ &= \ \asr(k-1)  \qquad \text{for every} \ \ \Cc \in \Par(S). \label{eq:ME2} \tag{ME2}
	\end{align}
\end{thm}

%\smallskip
%We say that weight function  $\ap: X \to \rrs$  is \defn{uniform} if
% $\ap(x)=\ap(y)$  for all $x,y\in X$.







%\nin
Condition~\eqref{eq:ME1} says that the  
$(k-1)$-continuation number is achieved on all independent sets 
$S\in \cI_{k-1}$, while
condition~\eqref{eq:ME2} is saying that all parallel classes
 $\Cc \in \Par(S)$  have the same size.
In contrast to
Theorem~\ref{t:matroids-equality},
there is a  a rich family of examples satisfying equality conditions in Theorem~\ref{t:matroids-equality-Par}, such as \emph{paving matroids} and \emph{Steiner system matroids}.
In particular,  we obtain the equality conditions
for the  special case of graphical matroids.

%\smallskip

\begin{prop}[{\defng{Equality for graphical matroids}}{}]\label{p:graphical-equality}
%	Let  $G=(V,E)$  be a simple connected graph on  $|V|=\sn$  vertices, and let
%	 $\aIr(k)$  be the number of  forests with $k$ edges.  Then
%	\begin{equation}\label{eq:prop-graphical-equality}
%		\frac{\aIr(\sn-2)^2}{\aIr(\sn-3) \. \cdot \. \aIr(\sn-1)}  \  \ge \  \frac{3}{2} \. \bigg(1\. + \. \frac{1}{\sn-2} \bigg).
%	\end{equation}
		Equality occurs in \eqref{eq:prop-graphical-equality} \, \underline{if and only if} \, $G$  is an  $\sn$-cycle.
	% Furthermore, for all  $1\le k \le \sn-3$,
	% the refined log-concavity inequality~\eqref{eq:matroid-Par} for  $\Mf_G$  is always strict.
\end{prop}





\section{Log-concavity for morphisms}\label{ss:intro-morphism}
%
For a matroid  $\Mf = (X,\Ic)$, the \defn{rank function}  $\rkm \colon 2^{X} \to \rrs$   is defined by
\[
\rkm(S) \ := \ \max  \. \big\{   |A| \,: \, A \subseteq  S, \, A \in \Ic  \big\}.
\]
%the maximum cardinality of an independent set contained in~$S$.
Note that  $\rk(\Mf) = \rkm(X)$.
Let  $\Mf = (X,\Ic)$  and  $\Nf = (Y,\Jc)$  be two
matroids with rank functions~$\rkm$ and~$\rkn$, respectively.
Let  $\Mor \colon X\to Y$  be a function that satisfies
%
\begin{equation}\label{eq:morphisms-rk}
	\rkn\big(\Mor(T)\big) \, - \,  \rkn\big(\Mor(S)\big)  \  \leq \  \rkm(T) \, - \,  \rkm(S)
	\quad \text{ for every } \ S \subseteq T \subseteq X.
\end{equation}
%
In this case we say that  $\Phi \colon \Mf \to \Nf$  is a \defna{morphism}
of matroids.
%
A subset  $S \in \Ic$  is said to be a \defn{basis}
of  $\Mor$  if  $\rkn(\Mor(S))=\rk(\Nf)$.  In other words,
$S$ is contained in a basis of  $\Mf$, and  $\Mor(S)$ 
contains a basis of $\Nf$.
	The notion of morphism of matroids generalizes many classical notions in combinatorics
such as \emph{graph coloring}, \emph{graph embeddings}, \emph{graph homomorphism},
\emph{matroid quotients}, and are a special case of the \emph{induced matroids}.
We refer to~\cite{EH} for a detailed overview and further references.



%\subsection{Mason's inequality for morphisms}
Denote by $\Bc$ the set of bases
of  $\Mor \colon \Mf\to \Nf$,   let  $\Bc_k:=\Bc\cap\Ic_k$, and let  $\aB(k):=|\Bc(k)|$.
The following log-concave inequality is due to Eur and Huh~\cite{EH}.

%\smallskip

\begin{thm}[{\defng{Log-concavity for morphisms}~\cite[Theorem~1.3]{EH}}]\label{t:morphisms-EH}
	Let  $\Mf = (X,\Ic)$  and  $\Nf = (Y,\Jc)$  be matroids, let  $n:=|X|$, and let
	 $\Mor \colon \Mf \to \Nf$ be a morphism of matroids, and let \ $1\le k <\rk(\Mf)$. Then:
	\begin{equation}\label{eq:morphism-EH}
		\aBr(k)^2  \ \geq \
		\left(1  +  \frac{1}{k}\right)  \left(1  + \. \frac{1}{n-k}\right) \, \aBr(k-1) \, \aBr(k+1).
	\end{equation}
\end{thm}
%\smallskip
%\nin
Note that when  $Y=\{y\}$  and  $\Nf=(Y,\emp)$  is
defined by  $\rkn(y)= 0$, we have condition~\eqref{eq:morphisms-rk} holds trivially
and  $\Bc=\Ic$.  Thus, the theorem generalizes Theorem~\ref{t:matroids-BH} to the
morphism of matroids setting.


Equality conditions  for Theorem~\ref{t:morphisms-EH} was posed as an open problem in \cite[Question~5.7]{MNY}, which  we resolved through   the following theorem.
%\smallskip

\begin{thm}[{\defng{Equality for morphisms}}]
	\label{t:morphisms-EH-equality}
%	Let  $\Mf = (X,\Ic)$  and  $\Nf = (Y,\Jc)$  be matroids, let  $n:=|X|$, and let
%	 $\Mor: \Mf \to \Nf$ be a morphism of matroids.  In addition, let  $\apr: X \to \rrs$ 
%	be a positive weight function, and let \ $1\le k <\rk(\Mf)$. Suppose  $\aBr_\apr(k)>0$.
%	Then:
%	\begin{equation}\label{eq:morphism-EH-equality}
%		\aBr_\apr(k)^2  \ = \
%		\left(1+ \frac{1}{k}\right)  \left(1+ \frac{1}{n-k}\right) \, \aBr_\apr(k-1) \, \aBr_\apr(k+1).
%	\end{equation}
Equality occurs in \eqref{eq:morphism-EH}
	\underline{if and only if} \, $\girth(\Mf)>k+1$
	and \. $\rkn\big(\Mor(S)\big)=\rk(\Nf)$  for all  $S\in \Ic_{k-1}$.
\end{thm}

%\smallskip

We now present a refinement of log-concavity for morphisms.
Recall the equivalence relation \. ``$\sim_S$'' \. on the set \.
$\Cnt(S)\subseteq X \setminus S$ \. of continuations of $S \in \Ic$,
see~\eqref{eq:matroid-cont-def}.  Similarly, recall the set  $\Par(S)$ 
of parallel classes of~$S$, see~\eqref{eq:matroid-p-def}.  For every \. $1\le k \leq \rk(\Mf)$, let
\[
\aP(k) \ := \ \max \. \bigl\{   \big|\Par(S)\big|  \. : \.  S \in \Bc_k  \bigr\},
\]
the maximum of the number of parallel classes of bases of morphism~$\Mor$ of size~$k$.

%\smallskip

\begin{thm}[{\defng{Refined log-concavity for morphisms}}]\label{t:morphisms-refined}
	Let  $\Mf = (X,\Ic)$  and  $\Nf = (Y,\Jc)$  be matroids,  let
	 $\Mor \colon \Mf \to \Nf$ be a morphism of matroids, and let \. $1\le k <\rk(\Mf)$.  Then:
	\begin{equation}\label{eq:morphism-refined}
		\aBr(k)^2  \ \geq \
		\left(1 +  \frac{1}{k}\right) \left(1 + \. \frac{1}{\aPr(k-1)\. -\. 1}\right) \,  \aBr(k-1) \, \aBr(k+1).
	\end{equation}
\end{thm}

%\smallskip

%\nin
As before, since \. $\aP(k-1) \leq n-k+1$, the theorem is an extension of
Theorem~\ref{t:morphisms-EH}.
We also obtain
 equality conditions for \eqref{eq:morphism-refined}, which we  present below.

%\smallskip

\begin{thm}[{\defng{Refined equality for morphisms}}]
	\label{t:morphisms-Par-equality}
	%
	Equality occurs in \eqref{eq:morphism-refined} \,
	\underline{if and only if} \, there exists  $\asr(k-1)>0$, such that for every  $S \in \Ic_{k-1}$  we have:
	\begin{align}
		\bigl|\Par_{S}\bigr| \ &= \  \aPr(k-1),   \label{eq:MorEqu1} \tag{MME1}\\
		|\Cc| \ &= \ \asr(k-1)  \qquad \text{for every} \ \ \Cc \in \Par(S),
		\ \text{and}\label{eq:MorEqu2} \tag{MME2}\\
		\rkn\bigl(\Mor(S)\bigr)  & =  \rk(\Nf).  \label{eq:MorEqu3} \tag{MME3}
	\end{align}
\end{thm}


\section{Discrete polymatroids}\label{ss:intro-poly}
%
A \defna{discrete polymatroid},\footnote{\defng{Discrete polymatroids} 
	are related but should not to be confused with \defng{polymatroids}, which is a family of
	convex polytopes, see, \textit{e.g.},~\cite[$\S$44]{Schr}.}
also called \defna{integral polymatroid},
 \. $\cD$  is a pair  $([n],\Jc)$  of a
ground set  $[n]:=\{1,\ldots,n\}$  and a nonempty finite collection 
$\Jc$   of integer points  $\ba = (a_1,\ldots,a_n)\in \nn^n$ 
that satisfy the following:
\begin{itemize}
	%	\item \, $\zero \in \Ic$,
	\item (\defng{hereditary property}) \. $\ba \in \Ic$, \. $\bb \in \nn^{n}$ such that $\bb \leqslant \ba \Rightarrow \bb \in \Ic$, and
	\item (\defng{exchange property}) \. $\ba, \bb \in \Ic$, $|\ba|< |\bb|$ implies there exists $i \in [n]$ such that $a_i < b_i$ and $\ba+\bbe_i \in \Jc$.
\end{itemize}
%
Here  $\bb \leqslant \ba$  is a componentwise inequality, \. $|\ba|:= a_1+\ldots+a_n$, and 
$\{\bbe_1,\ldots,\bbe_n\}$  is a standard linear basis in~$\rr^n$.
%
When  $\Jc\subseteq \{0,1\}^n$, discrete polymatroid  $\cD$  is a matroid.
One can think of a discrete polymatroid as a set system where multisets are allowed,
so we refer to~$\Jc$ as \defn{independent multisets} and to  $|\ba|$  as \defn{size}
of the multiset~$\ba$.
We refer to~\cite{HH} for the history and algebraic motivation.


%
%The role of bases in discrete polymatroids
%is played by \emph{maximal elements} with respect to the order~``$\leqslant$''; they are called
%\defng{M-convex sets} in~\cite[$\S$2]{BH}.

%\smallskip

 Define  $\rk(\cD):= \max \{ |\ba|\.:\. \ba\in\Jc\.\}$.
For  $0\le k \le \rk(\cD)$,
denote by  $\Jcm_k:=\{\ba\in \Jc\.:\.|\ba|=k\}$  the subcollection
of independent multisets of size~$k$.
Let  $\ap \colon  [n] \to \rrs$  be a positive \defn{weight function} on~$[n]$.  We extend weight function~$\ap$
to all  $\ba \in \Jc$  as follows:
\[  \ap(\ba) \ :=   \. \ap(1)^{a_1} \. \cdots \. \ap(n)^{a_n}.
\]
For every  $0\le k \le \rk(\cD)$, define
\[
\aJ_\ap(k) \ := \  \sum_{\ba \in  \Jcm_k} \,   \frac{\ap(\ba)}{\ba!} \ , \quad \text{where} \quad
\ba! \, := \, a_1! \, \cdots  \, a_n!.
\]
The following log-concavity  follows easily  from the results in~\cite{BH},
which was stated in a somewhat different form.

%\smallskip

%Indeed, we need Theorem 3.10 combined with taking derivatives and
%limits in proof of Theorem~4.14, and where Theorem~2.10 is
%substituted with Theorem~2.30 (all in~\cite{BH}).  The details are straightforward.
\begin{thm}[{\defng{Log-concavity for polymatroids}~\cite[Theorem~3.10~$(4)\Leftrightarrow(7)$]{BH}}]
	\label{t:polymatroids-BH}
	Let  $\cD=([n],\Jc)$  be a discrete polymatroid, and
	let   $\apr \colon  [n] \to \rrs$  be a positive weight function.
	For every  $1\le k < \rk(\Mf)$, we have:
	\begin{equation}\label{eq:polymatroid-BH}
		\rJ_\apr(k)^2 \ \ge  \left(1\. + \. \frac{1}{k}\right)  \,\.   \rJ_\apr(k-1)  \   \rJ_\apr(k+1).
	\end{equation}
\end{thm}

%\smallskip


 Equality conditions for \eqref{eq:polymatroid-BH} was previously unknown,
and we established them through the following theorem.
A discrete polymatroid  $\cD=([n],\Jc)$  is called \defn{nondegenerate} if \.
$\eb_i \in \Jc$ \. for every  $i \in [n]$.
Note that  every discrete polymatroid can be made to be nondegenerate by restricting $[n]$ to  integers $i$ satisfying  \. $\eb_i \in \Jc$.
Define \.
$\polygirth(\cD) \. := \. \min  \bigl\{k\.:\.|\Jc_k|<\binom{n+k-1}{k-1}\bigr\}$.
Observe that  $\ba\in \Jc$  for all  $\ba\in \nn^k$, \. $|\ba|<\polygirth(\cD)$.
Note that the \defn{polygirth} of a discrete polymatroid does not coincide with the girth
of a matroid. In fact, $\polygirth(\cD)=2$  when  $\cD$  is a matroid with
more than one element.

%\smallskip

\begin{thm}[{\defng{Equality for polymatroids}}]
	\label{t:polymatroids-BH-equality}
	Let  $\cD$  be a nondegenerate discrete polymatroid.
	Then equality occurs in \eqref{eq:polymatroid-BH} \,
	\underline{if and only if} \, $\polygirth(\cD)  >  (k+1)$.
\end{thm}




%We now give a common generalization of Theorem~\ref{t:matroids-Par-weighted} and
%Theorem~\ref{t:polymatroids-BH}.
We now present a refinement of log-concavity for discrete polymatroids.
 Fix  $\rnr \in [0,1]$. For every $\ba \in \Jc$
let
\[
\pi(\ba)\, := \ \sum_{i=1}^{n} \. \binom{a_i}{2}.
\]
For every  $0\le k \le \rk(\cD)$, define
$$\aJ_{\ap,\rnr}(k) \ := \  \sum_{\ba \in  \Jcm_k} \, \rnr^{\pi(\ba)}\,\.  \frac{\ap(\ba)}{\ba!}\,.
$$
Note that  $\binom{a}{2}=0$  for $a\in \{0,1\}$, so  $\pi(\ba)=0$  for all independent
sets  $\ba\in \cI$  in a matroid.


%\smallskip

For an independent multiset  $\ba\in \Jc$  of a discrete polymatroid  $\cD=([n],\Jc)$,
denote by
%
\begin{equation}\label{eq:multi-Cont-def}
	\Cnt(\ba) \, := \bigl\{i \in [n]~:~\ba +\bbe_i \in \Jc\bigr\}.
\end{equation}
%
the set of \defnb{continuations} of~$\ba$. For all  $i, j\in \Cnt(\ba)$, we write 
$i\sim_\ba j$  when  $\ba+\bbe_i+\bbe_j \notin\Jc$  or  $i=j$.  This is an
equivalence relation again.
We call an equivalence class of the relation
 $\sim_\ba$  a \defn{parallel class} of~$\ba$,
and we denote by $\Par(\ba)$ the set of parallel classes of~$\ba$.

For every  $0\le k< \rk(\cD)$, define the  \defnb{$k$-continuation number} of a
discrete polymatroid~$\cD$ as
the maximal  number of parallel classes of independent multisets of size~$k$:
\begin{equation}\label{eq:multi-p-def}
	\aP(k) \ := \ \max   \bigl\{ \.  \big|\Par(\ba)\big|  \ : \  \ba \in \Jcm_k \. \bigr\}.
\end{equation}
%
For matroids, this is the same notion as defined above.

%\smallskip

\begin{thm}[{\defng{Refined log-concavity for polymatroids}}]
	\label{t:polymatroids-Par}
	Let  $\cD=([n],\Jc)$  be a discrete polymatroid, and
	let   $\apr \colon  [n] \to \rrs$  be a positive weight function.
	For every  $\rnr \in [0,1]$  and  $1\le k < \rk(\Mf)$, we have:
	\begin{equation}\label{eq:polymatroid-Par}
		\rJ_{\apr,\rnr}(k)^2 \ \ge  \left(1\. + \. \frac{1}{k}\right)  \,
		\left(1\. + \, \frac{1-\rnr}{\aPr(k-1)   -  1  + \rnr}\right) \,\.   \rJ_{\apr,\rnr}(k-1)  \   \rJ_{\apr,\rnr}(k+1).
	\end{equation}
\end{thm}

%\smallskip

%\nin
When  $\rnr=1$, this gives Theorem~\ref{t:polymatroids-BH}.
When  $\cD$  is a matroid,  $\rnr=0$, and $\apr$ is an uniform weight,
this gives Theorem~\ref{t:matroids-Par}.
For general discrete polymatroids~$\cD$ and  $0<\rnr<1$, this
is a stronger result.

To get the equality conditions for~\eqref{eq:polymatroid-Par}, we separate
the cases  $\rnr=0$, \. $0<\rnr<1$, and  $\rnr=1$. The case  $\rnr=0$  coincides
with equality conditions for matroids given in Theorem~\ref{t:matroids-equality-Par}; the case  $\rnr=1$  coincides
with equality conditions for polymatroids given in Theorem~\ref{t:polymatroids-BH-equality}; and the case  $0<\rnr<1$  is contained in the following lemma.

%\smallskip


\begin{thm}[{\defng{Refined equality for polymatroids} $0<\rnr<1$  case}]
	\label{t:polymatroids-Par-equality}
	Let  $\cD$  be a nondegenerate discrete polymatroid, and let  $0<\rnr<1$.  Then equality occurs in \eqref{eq:polymatroid-Par} \,
	\underline{if and only if} \, $k=1$, \. $\polygirth(\cD)  >  2$, and  $\apr$  is uniform.
\end{thm}

%\smallskip
%\nin
One can view the dearth of nontrivial examples in Theorem~\ref{t:polymatroids-Par-equality} as suggesting that the bound in Theorem~\ref{t:polymatroids-Par}
can be further improved for  $\rnr>0$.  This is based on the reasoning
that Theorem~\ref{t:matroids-Par} sharply improves over Theorem~\ref{t:matroids-BH}
because there are only trivial equality conditions for the latter
(see Theorem~\ref{t:matroids-equality}), when compared with rich equality
conditions for the former (see Theorem~\ref{t:matroids-equality-Par}).


\section{Poset antimatroids}

Let  $X$  be finite set we call \defng{letters}, and let  $X^\ast$  be a set of
\defng{finite words} in the alphabet~$X$.  A \defn{language over~$X$} 
is a nonempty finite subset  $\cL \ssu X^\ast$.
A word is called \defn{simple} if it contains each letter at most once;
we consider only simple words from this point on.  We write  $x\in \al$ 
if word  $\al\in\cL$  contains letter~$x$. Finally, let
 $|\al|$  be the length of the word, and denote
 $\cL_k := \bigl\{\al\in \cL\.:\. |\al|=k\bigr\}$.

%\smallskip

A pair  $\Af = (X,\cL)$  is an \defna{antimatroid}, if the language
 $\cL\ssu X^\ast$  satisfies:
%
\begin{itemize}
	\item (\defng{nondegenerate property})  \,
	every  \. $x\in X$ \. is contained
	in at least one \. $\al\in \cL$,
	%
	\item (\defng{normal property}) \, every  \. $\al\in \cL$ \. is simple,
	%
	\item (\defng{hereditary property}) \, $\al\be\in \cL$  \ $\Rightarrow$ \ $\al\in \cL$, \. and
	%
	\item (\defng{exchange property}) \,  $x\in \al$, \.  $x\notin\beta$,  \. and \.   $\al,  \beta \in \cL$ implies there exists $ y \in \al$ such that $\beta y\in \cL$.
	%
\end{itemize}

%\smallskip


%\nin
Note that for every antimatroid  $\Af = (X,\cL)$, it follows from the exchange
property that
$$
\rk(\Af) \. := \. \max\{ |\al|\.:\.\al\in\cL\} \. = \. |X|.
$$
Antimatroids is a subclass of greedoids named after the \emph{anti-exchange property},
which is a key axiom in their definition via set systems.
 We refer to~\cite{KLS} for the history and geometric motivation.




In this extended abstract we use only one class of antimatroids
which we now define.
Let  $\cP=(X,\prec)$  be a poset on  $|X|=n$  elements.
A simple word  $\al\in X^\ast$  is called \defn{feasible} if  $\al$  satisfies:
\begin{itemize}
	\item (\defng{poset property}) \, if  $\alpha$ contains $x\in X$ and $y \prec x$,
	then letter $y$ occurs before letter $x$ in $\alpha$.
\end{itemize}
%
A \defna{poset antimatroid} \. $\Af_\cP=(X,\cL)$  is defined
by the language  $\cL$  of all feasible words in~$X$.
% The
%exchange property is satisfied because one can always take $y$
%to be the smallest letter (w.r.t.\ order~$\prec$) that is not in~$\be$.

%\smallskip

Let  $\ap \colon X \to \rrs$  be a positive weight function on~$X$.
Denote by  $\Cov(x) := \{y\in X: x \lcov  y\}$  the set of elements
which cover~$x$.  % Define:
%
We assume the weight function  $\ap$  satisfies the following
%
 (\defng{cover monotonicity property}):
%
\begin{equation}\label{eq:CM}\tag{CM}
	\ap(x)  \, \geq \,  \sum_{y \in \Cov(x)} \. \ap(y)\., \quad\text{for all \ $x\in X$.}
\end{equation}

%\smallskip

%\nin
Note that when~\eqref{eq:CM} is equality for all $x\in X$, we have:
\begin{equation}\label{eq:ap Hasse}
	\ap(x) \, = \, \text{number of maximal chains in~$\cP$ starting at~$x$.}
\end{equation}

%\medskip

% We extend $\ap$  to all feasible words  $\al \in \cL$  by
% \[  \ap(\alpha)  \, := \,  \prod_{x \in \alpha} \. \ap(x).
% \]

For all  $\al \in \cL$  and  $0\le k \le n$, let
$$\aL_\ap(k) \, := \, \sum_{\al\in \cL_k} \. \ap(\al)\,, \quad
\text{where} \quad  \ap(\alpha)  \, := \,  \prod_{x \in \alpha} \. \ap(x).
$$

%\smallskip

\begin{thm}[{\defng{Log-concavity for poset antimatroids}}]\label{t:antimatroids}
	Let  $\cP=(X,\prec)$  be a poset on  $|X|=n$  elements,
	and let  $\cA_\cP=(X,\cL)$  be the corresponding poset antimatroid.
	Let  $\apr \colon X \to \rrs$  be a positive  weight function
	which satisfies~\eqref{eq:CM}.  Then, for every integer  $1\le k < n$,
	we have:
	\begin{equation}\label{eq:antimatroid-LC}
		\aLr_\ap(k)^2 \, \ge \, \aLr_\ap(k-1)  \cdot  \aLr_\ap(k+1).
	\end{equation}
\end{thm}

%\smallskip


	%\nin
 	Note  that the weight function condition~\eqref{eq:CM} is necessary,
	as there are examples for which log-concavity fails to hold for uniform weight, see \cite[Example~1.27]{CP} in the full version of this extended abstract.
% We also obtained
%	equality conditions for \eqref{eq:greedoid-LC}, which we omit as it is too technical to state here.



%\medskip


%\section{Interval greedoids}
%Let  $X$  be finite set of letters, and let  $\cL\ssu X^\ast$ \ts be a
%nonempty language over~$X$.  A pair \ts $\Gf=(X,\cL)$ \ts is a \defna{greedoid}, if the
%language~$\cL$ satisfies:
%
%\begin{itemize}
%%	\item  (\defng{nondegenerate property})  \, empty word \ts $\varnothing$ \ts is in $\Lc$,
%%	%
%	\item (\defng{normal property}) \, every  \ts $\al\in \cL$ \ts is simple,	
%	%
%	\item (\defng{hereditary property}) \, $\al\be\in \cL$  \ $\Rightarrow$ \ $\al\in \cL$, and
%	%
%	\item (\defng{exchange property}) \,  $\al, \be\in \cL$ \. s.t.\  \. $|\alpha| > |\beta|$
%	\ $\Rightarrow$ \ $\exists\ts x\in \al$ \, s.t.\  \. $\beta x \in \Lc$\ts.
%\end{itemize}
%
%%\smallskip
%
%\nin
%
%
%
%%\smallskip
%
%Greedoid \ts $\Gf=(X,\cL)$ \ts is called \defna{interval} if the
%language~$\cL$ also satisfies:
%\begin{itemize}
%	\item (\defng{interval property}) \,  $\alpha,\beta,\gamma \in X^*$, \. $x \in X$ \ts \.
%.t.\  \. $\alpha x, \ts \alpha \beta \gamma x \in \Lc$
%	\ \, $\Rightarrow$ \ \,$\alpha \beta x \in \Lc$\ts.
%\end{itemize}
%It is well known and easy to see that matroids are interval greedoids.
%
%%\smallskip
%
%Let \. $\aq: \cL \to \rrs$ \. be a positive weight function.  Let
%$$
%\aL_\aq(k) \, = \, \sum_{\al\in \cL_k} \. \aq(\al)\ts.
%$$
%We established  log-concavity for interval greedoids in the following theorem under the assumption that $\aq$ is a  \ts \defna{$k$-admissible} \ts
%weight function.
%We omit the definition of $k$-admissibility  as it  is much too technical to state here.
%
%%\smallskip
%
%\begin{thm}[{\rm \defng{Log-concavity for interval greedoids}}{}]
%	\label{t:greedoid}
%	Let \ts $\Gf=(X,\cL)$ \ts be an interval greedoid, let \ts $1\le k <\rk(\Gf)$,
%	and let \. $\aqr:\cL \to \rrs$ \. be a \ts {$k$-admissible} \ts weight function.
%	Then:
%	\begin{equation}\label{eq:greedoid-LC}
%		\aLr_\aqr(k)^2 \ \geq \   \aLr_\aqr(k-1) \. \cdot \. \aLr_\aqr(k+1)\ts.
%	\end{equation}
%\end{thm}
%
%%\smallskip
%
%\nin This theorem implies all previous inequalities for
%matroids~(Theorem~\ref{t:matroids-Par}) and  polymatroids~(Theorem~\ref{t:polymatroids-Par}).
%We also obtained
%equality conditions for \eqref{eq:greedoid-LC},
% which we use to obtain the
%equality conditions for matroids~(Theorem~\ref{t:matroids-equality-Par})
%and  polymatroids~(Theorem~\ref{t:polymatroids-Par-equality}).
%We omit the conditions  as it is much too teachnical to state here.



\section{Linear extensions}

Let  $\Pf:=(X,\prec)$  be a poset on  $n:=|X|$  elements.
A \defna{linear extension} of $\Pf$ is a bijection \. $L \colon X \to \{1,\ldots, n\}$, such that
\. $L(x) < L(y)$ \. for all \. $x \prec y$.
%Linear extensions of a finite poset  $\cP$  are in an obvious bijection with
%maximal chains in the lattice  $\LL(\cP)$  of lower order ideals of~$\cP$.
%Lattice  $\LL(\cP)$  is always distributive, and by
%\emph{Birkhoff's representation theorem} (see e.g.\ \cite[Theorem~3.4.1]{EC}),
%every finite distributive lattice can be obtained that way.
We refer to
\cite{BrW,Tro} for definitions and standard results on posets and linear
extensions.




Fix an element  $\zf\in X$.
Denote by  $\Ec := \Ec(P)$  the set of linear extensions
of~$\Pf$, let \. $\Ec_k :=  \{L \in \Ec\.:\.L(\zf)=k\}$.
The following inequality  was originally conjectured by
Chung, Fishburn and Graham in~\cite{CFG}, extending an earlier unimodality
conjecture by R.~Rivest (unpublished).


%\smallskip

\begin{thm}[{\defng{Stanley inequality}~\cite[Theorem~3.1]{Sta}}]\label{t:Sta}
	Let   $\cP=(X,\prec)$  be a poset with  $|X|=n$  elements, and let  $\zf\in X$.
	Denote by  $\aNr(k):=|\Ec_k|$  the number of linear extensions  $L\in \Ec(P)$,
    such that  $L(\zf)=k$.  Then, for every \. $1< k < n$, we have:
	\begin{equation}\label{eq:Sta}
		\aNr(k)^2 \,\. \ge \,\. \aNr(k-1) \.\cdot \.  \aNr(k+1).
	\end{equation}
\end{thm}

%\smallskip

%\nin
The proof in~\cite{Sta} is a simple
application of the \emph{Alexandrov--Fenchel inequality}.
In the same paper, Stanley asked for equality conditions for \eqref{eq:Sta},
which were recently obtained by Shenfeld and van Handel.
Denote by  $f(x) := \bigl|\{y \in X\,{}:\,{}y\prec x\}\bigr|$  and 
$g(x) := \bigl|\{y \in X\,{}:\,{}y\succ x\}\bigr|$  the sizes of lower
and upper ideals of  $x\in X$, respectively, excluding the element~$x$.

%\smallskip

\begin{thm}[{\defng{Equality condition for Stanley inequality}~\cite[Theorem~15.3]{SvH}}]\label{t:Sta-equality}
%	Let   $\Pf=(X,\prec)$  be a poset with  $|X|=n$  elements.  Let  $\zf\in X$  and
%	let  $\aNr(k)$  be the number of linear extensions  $L\in \Ec(P)$,
%such that  $L(\zf)=k$.
Suppose that  $\aNr(k)>0$.
	Then \. \underline{the following are equivalent}:
	%
	\begin{enumerate}
		[{label=\textnormal{({\alph*})},
			ref=\textnormal{\alph*}}]
		%
		\item \ $\aNr(k)^2 \. = \. \aNr(k-1) \. \cdot \. \aNr(k+1)$,
		\item \ $\aNr(k+1) \. = \. \aNr(k) \. = \. \aNr(k-1)$,
		\item \ we have \. $f(x)>k$ \. for all \. $x\succ \zf$, \. and \. $g(x)>n-k+1$ \. for all \. $x\prec \zf$.
	\end{enumerate}
\end{thm}


The proof in \cite{SvH} used a sophisticated geometric analysis to prove
equality conditions of the Alexandrov--Fenchel inequality for convex polytopes.






We now give a weighted generalization of these results, which we proved combinatorially.  Let  $\ap \colon X \to \rrs$ 
be a positive weight function on~$X$.  We say that  $\ap$  is \defn{order-reversing}
if it satisfies
\begin{equation}\label{eq:Rev}\tag{Rev}
	x \. \preccurlyeq\. y \quad \Rightarrow 	\quad  \ap(x) \,  \geq \, \ap(y).
\end{equation}
Fix  $\zf\in X$, as above.   Define \. $\ap \colon \Ec\to \rrs$ \. by
\begin{equation}\label{eq:sta-def-omega-weighted}
	\ap(L)  \ := \ \prod_{x \, : \, L(x) < L(\zf)} \. \ap(x),
\end{equation}
and let
\begin{equation}\label{eq:sta-def-N-weighted}
	\aN_\ap(k) \, := \, \sum_{L \in \Ec_k} \. \ap(L)\., \quad \text{for all} \ \ 1\le k \le n.
\end{equation}


%\smallskip

\begin{thm}[{\defng{Weighted Stanley inequality}}]\label{t:Sta-weighted}
	Let   $\cP=(X,\prec)$  be a poset with  $|X|=n$  elements, and
	let  $\ap\colon X \to \rrs$  be a positive order-reversing weight function.
	Fix an element  $\zf\in X$.
	Then, for every \. $1< k < n$, we have:
	\begin{equation}\label{eq:Sta-weighted}
		\aNr_\apr(k)^2 \,\. \ge \,\. \aNr_\apr(k-1) \.\cdot \. \aNr_\apr(k+1),
	\end{equation}
	where  $\aNr_\apr(k)$  is defined by~\eqref{eq:sta-def-N-weighted}.
\end{thm}


%\smallskip
%\nin
Prior to our work, no direct combinatorial proof of Stanley's inequality was known
in full generality, although \cite{CFG} gives a simple proof for posets
of width two (see also~\cite{CPP2}).  Most recently, the authors and
Panova obtained a $q$- and multivariate analogues of Stanley's
inequality for posets of width two~\cite{CPP2}.  These notions are specific to
the width two case and are incompatible with the weighted analogue above.

The equality conditions for  \eqref{eq:sta-def-omega-weighted} is a little more subtle and needs
the following  \defng{$(\as,k)$-cohesiveness property}:
%
\begin{equation}\label{eq:cohesive}\tag{Coh}
	\ap\bigl(L^{-1}(k-1)\bigr) \. = \. \ap\bigl(L^{-1}(k+1)\bigr) \. = \. \as, \quad
	\text{for all} \ \ L \in \Ec_k\,.
\end{equation}

%\smallskip

%Note that~\eqref{eq:cohesive} can hold for non-uniform weight functions~$\ap$,
%for example for  $\Pf = \rA_{k+1} \oplus \rC_{n-k-1}$, i.e.\ the \emph{linear sum} of an
%antichain on which $\ap$ is uniform and a chain on which $\ap$ can be non-uniform.
%In fact, if $\zf$ is an element in~$\rA_{k+1}$, we can have  $\ap(\zf)$ 
%different from the rest of the antichain.
%
%%\smallskip

\begin{thm}[{\defng{Equality condition for weighted Stanley inequality}}]\label{t:Sta-equality-weighted}
 Suppose that  $\aNr_\apr(k)>0$.
	Then \. \underline{the following are equivalent}:
	\begin{enumerate}
		[{label=\textnormal{({\alph*})},
			ref=\textnormal{\alph*}}]
		%
		\item $\aNr_\apr(k)^2 \. = \. \aNr_\apr(k-1) \. \cdot \.\aNr_\apr(k+1)$,
		\item there exists \. $\asr=\asr(k,\zf) >0$ such that  % we have~\eqref{eq:cohesive} and
		\[\aNr_\apr(k+1) \. = \. \asr \. \aNr_\apr(k) \. = \. \asr^2 \. \aNr_\apr(k-1),
		\]
		\item there exists $\asr=\asr(k,\zf) >0$ such that
		$f(x)>k$ for all  $x\succ \zf$, $g(x)>n-k+1$ for all $x\prec \zf$, and~\eqref{eq:cohesive}.
	\end{enumerate}
\end{thm}


\section{Proof ideas} \label{ss:intro-proof}
%
Although we prove multiple results, the proof of each log-concavity
inequality uses the same approach and technology, so we refer to
it as ``the proof''.

At the first level, the proof is an inductive argument proving a stronger
claim about eigenvalues of certain matrices associated with the posets.
The induction is not over posets of smaller size, but over other matrices
which can in fact be larger, but correspond to certain parameters decreasing
as we go along.  The claim then reduces to the base of induction,
which is the only part of the proof requiring a computation.  The latter
involves checking eigenvalues of explicitly written small matrices, making
the proof fully elementary.

Delving a little deeper, we set up a new type of structure which we call a 
\defna{combinatorial atlas}.  In the special case of matroids,
a combinatorial atlas   $\AA$  associated with a
matroid  $\Mf = (X,\Ic)$,  $|X|=n$,  is comprised of:
\begin{itemize}
	\item[$\circ$] Acyclic digraph \. $\Qf_\Mf=(X^*,\Ef)$,  with the unique source at the
	empty word  $\emp\in X^*$, and edges
	corresponding to
	multiplications by a letter: \ $
	\Ef  =  \bigl\{(\al,\al x)\.:\. \al,\al x\in X^*, \. x\in X\bigr\}$,
	
	\item[$\circ$] Each vertex  $\al \in X^*$  is associated with a pair 
	$(\bM_\al,\hb_\al)$,  where  $\bM_\al=\big(\aM_{i j}\big)$  is a nonnegative symmetric  $\ar\times \ar$  matrix,  $\hb_\al = (\ah_1,\ldots,\ah_\ar)$  is a nonnegative vector,
	and  $\ar=n+1$,
	
	\item[$\circ$]  Each edge  $(\al,\al x)\in \Ef$  is associated with a linear  transformation 
	$\bT_\al^{\langle x\rangle} \colon \rr^\ar\to \rr^\ar$.
\end{itemize}



%\nin
The key technical observation is that under certain conditions on the atlas,
we have every matrix
 $\bM:=\bM_\al$,   is
% \defn{Lorentzian}, i.e.\ the corresponding quadratic form is
\defn{hyperbolic}:
\begin{equation}\label{eq:Hyp-intro}
	\langle \vb, \bM \wb \rangle^2 \  \geq \  \langle \vb, \bM \vb \rangle  \langle \wb, \bM \wb \rangle \quad \text{for all $\vb, \wb \in \Rb^{\ar}$  such that $\langle \wb, \bM \wb \rangle > 0$}.   \tag{Hyp}
\end{equation}
%
Log-concavity inequalities now follow from~\eqref{eq:Hyp-intro} for the matrix
 $\bM_\emp$, by interpreting the inner products as numbers  $\aL_\aq(k)$,
 $\aL_\aq(k-1)$  and  $\aL_\aq(k+1)$, respectively.

We prove~\eqref{eq:Hyp-intro} by induction, reducing the claim for  $\bM_\al$ 
to that of  $\bM_{\al x}$ , for all  $x\in \Cnt(\al)$.
%
Proving~\eqref{eq:Hyp-intro}
for the base of induction required the eigenvalue interlacing argument.    This is
where our conditions for the weight function~$\ap$ appear in the calculation.
%
We also need a few other properties
of the atlas.  Notably, we require every matrix  $\bM_\al$  to be \defn{irreducible}
with respect to its support, but that is proved by a direct combinatorial argument.

%\smallskip

For other log-concavity inequalities in the paper, we consider similar atlas constructions
and similar claims.  For the equalities, we works backwards and observe that we need
equations~\eqref{eq:Hyp-intro} to be equalities.  These imply the local properties
which must hold for certain edges  $(\al,\al x)\in \Ef$.  Analyzing these properties
gives the equality conditions we present.

Although this can only be understood from the proofs, much of this work has been 
influenced by~\cite{SvH19} which gives a new proof of the Alexandrov--Fenchel 
inequalities.  The closest we come to~\cite{SvH19} is in the most technical part
of the paper on Stanley's inequality, but again the tools we employ are highly 
technical and go much beyond what can be described in the extended abstract.  
We refer to the full version~\cite{CP} for all the details.


%\acknowledgements{We are grateful to Marcelo Aguiar, Noga Alon, Nima Anari, Andy Berget,
%Olivier Bernardi, June Huh, Jeff Kahn, Joseph Kung, Jonathan Leake, Alejandro Morales,
%Yair Shenfeld, Hunter Spink and  Cynthia Vinzant for helpful discussions
%and remarks on the subject.
%%
%Special thanks to Ramon van Handel for many insightful
%comments  and
%for help with the references.
%%
%We are also thankful to Christian Ikenmeyer and Greta Panova
%for numerous interesting conversations on poset inequalities.
%}

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