\documentclass[finalversion]{FPSAC2023}
\articlenumber{69} %change submission to "revision", after revision

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[centertableaux]{ytableau}
\usepackage{tikz, tkz-euclide}
%\usepackage{PPPackage}
\usepackage[backend=bibtex]{biblatex}
\addbibresource{FPSAC2023.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Environments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Math operators %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\x}{{\bf x}}
\newcommand{\e}{{\operatorname{E}}}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\DPP}{{\mathbf{DPP}}}
\DeclareMathOperator{\ASM}{{\mathbf{ASM}}}
\newcommand{\W}{{\operatorname{W}}}

\newcommand{\nenwarrow}{\nwarrow \!\!\!\!\!\;\!\! \nearrow}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% other definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\defn}[1]{\emph{\color{cyan!70!black} #1}}
\newcommand{\red}[1]{{\color{red} #1}}

\definecolor{verylight-gray}{gray}{0.95}
\definecolor{light-gray}{gray}{0.9}

\pgfdeclaredecoration{arrows}{draw}{
	\state{draw}[width=\pgfdecoratedinputsegmentlength]{%
		\path [every arrow subpath/.try] \pgfextra{%
			\pgfpathmoveto{\pgfpointdecoratedinputsegmentfirst}%
			\pgfpathlineto{\pgfpointdecoratedinputsegmentlast}%
		};
}}
\tikzset{every arrow subpath/.style={->, draw, thick}}

\pgfdeclaredecoration{arrowsEven}{draw}{
	\state{draw}[width=\pgfdecoratedinputsegmentlength]{%
		\path [every arrow subpath even/.try] \pgfextra{%
			\pgfpathmoveto{\pgfpointdecoratedinputsegmentfirst}%
			\pgfpathlineto{\pgfpointdecoratedinputsegmentlast}%
		};
}}
\tikzset{every arrow subpath even/.style={->, draw, thick, cyan}}

\pgfdeclaredecoration{arrowsOdd}{draw}{
	\state{draw}[width=\pgfdecoratedinputsegmentlength]{%
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			\pgfpathmoveto{\pgfpointdecoratedinputsegmentfirst}%
			\pgfpathlineto{\pgfpointdecoratedinputsegmentlast}%
		};
}}
\tikzset{every arrow subpath odd/.style={->, draw, thick, magenta}}



\newcommand{\RightBoundary}[3]{
\begin{scope}[xshift= 1cm*cos(30)*(#1+#2) , yshift=1cm*#3+1cm*sin(30)*(#2-#1)]
\draw [fill=white] (0,0) -- ({1*cos(30)},{1*sin(30)}) -- ({1*cos(30)},{-1*sin(30)}) -- (0,-1) -- (0,0);
\end{scope}
}
\newcommand{\LeftBoundary}[3]{
\begin{scope}[xshift= 1cm*cos(30)*(#1+#2) , yshift=1cm*#3+1cm*sin(30)*(#2-#1)]
\draw (0,0)[fill=white] -- ({-1*cos(30)},{1*sin(30)}) -- ({-1*cos(30)},{-1*sin(30)}) -- (0,-1) -- (0,0);
\end{scope}
}
\newcommand{\TopBoundary}[3]{
\begin{scope}[xshift= 1cm*cos(30)*(#1+#2) , yshift=1cm*#3+1cm*sin(30)*(#2-#1)]
\draw (0,0)[fill=white] -- ({1*cos(30)},{1*sin(30)}) -- (0,1) -- ({-1*cos(30)},{1*sin(30)}) -- (0,0);
\end{scope}
}

	\newcounter{x}
	\newcounter{y}
	\newcounter{z}
	\newcounter{help}

\newcommand{\PlanePartition}[1]{
	\foreach \m [count=\y] in {#1}{		 								%\m = row, \y = number of the row
		\foreach \z [count=\x] in \m{									%\z = entry in the row, \x the position of the entry
			\setcounter{x}{\x}											%remembers the position of the current entry
			\ifnum \z > 0												% if the entry is positive
				\TopBoundary{\x}{-\y}{\z}								%draw its top
			\fi
			
			\ifnum \x > 1												%if we are not regarding the first entry of the row
				\ifnum \z < \thez										% and if the current entry is less than the previous			
					\setcounter{help}{\z}
					\addtocounter{help}{1}								%has to be done this way since an update (30.10.2019)
					\foreach \zz in {\thehelp,...,\thez}{					
						\RightBoundary{(\x -1)}{(-\y)}{\zz}			%draw the correct amount of right sides of the boxes in
					}														% the previous position.
				\fi
			\fi
			\ifnum \y > 1											%if we are at least in the second row
				\foreach \mm [count=\yy] in {#1}{					%\mm = another row and \yy is its number
				\ifnum \yy = \they									%if \mm is the row before \m							
				\foreach \zz [count=\xx] in \mm{					%let \zz be the entry in \mm and \xx its position
				\ifnum \xx = \x										%if the positions \xx and \x coincide
				\ifnum \zz > \z										% and the entry \zz is larger than \z
					\setcounter{help}{\zz}									%has to be done like this since the update
					\addtocounter{help}{-1}	
					\foreach \zzz in {\z,...,\thehelp}{
						\LeftBoundary{(\x -1)}{(1-\yy)}{\zzz}		% draw the correct amount of left sides for the boxes "\zz"
					}
				\fi
				\fi
				}
				\fi
				}
			\fi		
			\setcounter{z}{\z}										%remembers the previous entry	
		}										%done with all entries in row \m
		\ifnum \thez >0												%if the last entry is larger than 0
			\foreach \z in {1,...,\thez}{
				\RightBoundary{\thex}{-\y}{\z}						%draw the appropriate number of left sides of these boxes
			}
		\fi
		\setcounter{y}{\y}
	}											%done with row \m
	
	
	\foreach \mm [count=\yy] in {#1}{								%regards the left sides of the boxes in the last row
	\ifnum \yy = \they
	\foreach \zz [count=\xx] in \mm{
	\ifnum \zz >0
		\foreach \zzz in {1,...,\zz}{
			\LeftBoundary{(\xx -1)}{(1-\yy)}{(\zzz-1)}			%draw the appropriate number of left sides of the boxes
		}
	\fi
	}
	\fi
	}
	
}

\newcommand{\TilingBox}[3]{
	\foreach \a in {1,...,#2}{
		\foreach \b in {1,...,#3}{
			\RightBoundary{0}{-\a}{\b -1}
		}
	}
	\foreach \a in {1,...,#1}{
		\foreach \b in {1,...,#3}{
			\LeftBoundary{\a}{0}{\b}
		}
	}
	\foreach \a in {1,...,#1}{
		\foreach \b in {1,...,#2}{
			\TopBoundary{\a}{-\b}{0}
		}
	}
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% document setup %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title[ASMs and DPPs]{Alternating Sign Matrices and Descending Plane Partitions: a linear number of equivalent statistics}
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}
\author{
Ilse Fischer\thanks{\href{mailto:ilse.fischer@univie.ac.at}{ilse.fischer@univie.ac.at}. Ack.\ supp.\ from the Austrian Science Foundation FWF, P34931}\addressmark{1},
Hans H\"ongesberg\thanks{\href{mailto:hans.hoengesberg@univie.ac.at}{hans.hoengesberg@univie.ac.at}. 
Ack.\ supp.\ from the Austrian Science Foundation FWF, P34931}\addressmark{1}, \and
Florian Schreier-Aigner\thanks{\href{florian.schreier-aigner@univie.ac.at}{florian.schreier-aigner@univie.ac.at}.
Ack.\ supp.\ from the Austrian Science Foundation FWF, P34931 and J4387}\addressmark{1}
 }
\address{\addressmark{1}University of Vienna, Austria}
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

\abstract{There is the same number of alternating sign matrices as there is of cyclically symmetric lozenge tilings of a hexagon with a central triangular hole of size $2$, but finding an explicit bijection has been an open problem for about $40$ years now. This is even more surprising in the view of the fact that,  when restricting to vertically symmetric alternating sign matrices,
their number equals the number of such lozenge tilings that also exhibit an additional reflective symmetry.  To approach such bijections, we present generalizations of these results that involve a linear number of equidistributed statistics. Prior to this work, only a constant number of such statistics were known.}
%\resume{}

\keywords{alternating sign matrices, plane partitions, non-intersecting lattice paths, bijective proofs}



\begin{document}
\maketitle

\section{Introduction}
\label{sec:intro}
\input{intro.tex}

\section{Arrowed monotone triangles}
\label{sec:mts}
\input{mts.tex}

\section{On the relation between ASMs and DPPs}
\label{sec:asms}
\input{asms.tex}

\section{Reflective symmetry}
\label{sec:vsasms}
\input{vsasms.tex}


%% 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}
