An \defn{alternating sign matrix} (ASM) is a square matrix with entries in $\{0,\pm 1\}$ such that, in each row and each column,  $1$'s and $-1$'s alternate and sum to $1$. All $3 \times 3$ ASMs are given next.
$$
\footnotesize
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad
\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad
\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \quad
\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad
\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad
\begin{pmatrix} 0 & 1 & 0 \\ 1 & -1 & 1 \\ 0 & 1 & 0 \end{pmatrix}
$$
ASMs had been introduced by Robbins and Rumsey in the 1980s \cite{lambda} and 
they conjectured that the number of $n \times n$ ASMs is 
$\prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!}$,
which was proven after considerable effort by Zeilberger \cite{Zei96a} (in fact, he showed a more general result that includes an additional parameter). Soon after that, Kuperberg used methods from statistical physics to give another, shorter proof \cite{Kup96} (of the special case). 

Further observations caused considerable excitement in the combinatorics community, in particular $\prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!}$ appears also as the enumeration formula for \defn{descending plane partitions} (DPPs) with parts no greater than $n$ \cite{And79,MilRobRum83}, which are defined next. A \defn{strict partition} is a sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of positive integers with $\lambda_1 > \ldots > \lambda_n$, and the \defn{shifted Young diagram} of shape $\lambda$ has $\lambda_i$ cells in row $i$, where each row is indented by one cell to the right with respect to the previous row. A DPP 
is a filling of a shifted Young diagram with positive integers such that rows decrease weakly and columns decrease strictly, and
the first part of each row is greater than the length of its row and less than or equal to the length of the previous row. The DPPs with all entries less than or equal to $3$ are given next.
$$\footnotesize \emptyset \quad \begin{ytableau} 2 \end{ytableau} \quad  \begin{ytableau} 3 \end{ytableau} \quad \begin{ytableau} 3 &  1 \end{ytableau} \quad
\begin{ytableau} 3 & 2 \end{ytableau} \quad \begin{ytableau} 3 & 3 \end{ytableau} \quad \begin{ytableau} 3 & 3 \\ \none & 2 \end{ytableau}$$

There has been much effort to construct explicit bijections between ASMs and DPPs, but until recently without much progress. Very recently, a bijective proof \cite{cube} of an identity that implies the equinumerosity of ASMs and DPPs was established. 
%More precisely, if $\ASM_n$ denotes the set of $n \times n$ ASMs and $\DPP_n$ denotes the set of DPPs with parts no greater than $n$, a bijection between $\ASM_n \times \DPP_{n-1}$ and $\ASM_{n-1} \times \DPP_n$ has been constructed.
The bijection is quite involved, and uses signed sets and a generalization of the involution principle by Garsia and Milne  \cite{garsiamilne3}. It seems fair to say that a perfect understanding of these relations despite the many efforts is still missing.


The main purpose of this extended abstract is to shed light on the mysterious relation between ASMs and DPPs by introducing  
a pair of $n+3$ equidistributed statistics, one set of $n+3$ statistics on ASMs and the other set on DPPs. Finding such statistics is a natural approach to construct bijections. Prior to this work, a constant 
number of such statistics were known. The extended abstract is a  summary of our two preprints \cite{nASMDPP,FH22} and we refer the reader to these preprints for all the details.

The introduction of our statistics is at the cost of replacing both ASMs and DPPs by somewhat modified
objects, see \cref{sec:mts,sec:asms}, respectively, however the relation of these modified objects to ASMs and DPPs is easily established by combinatorial means. We are also able to obtain a result that is similar in vein for vertically symmetric ASMs and the corresponding DPP-objects (\cref{sec:vsasms}). As the results are also phrased in terms of \defn{non-intersecting lattice paths} on the DPP-side, we spend the rest of the introduction by presenting the lozenge tilings that correspond to DPPs which in turn are in easy bijective correspondence with the lattice paths.


DPPs with parts no greater than $n$ are known \cite{Kra06} to be in easy bijective correspondence with cyclically symmetric lozenge tilings of a hexagon with side lengths $n,n+2,n,n+2,n,n+2$ that have a central triangular hole of size $2$, see \cref{fig: lozenge tiling and DPP}. Such tilings can be encoded as families of non-intersecting lattice paths (\defn{$n$-DPP paths}), see \cref{fig: lozenge tiling and DPP}. More precisely, the \defn{$n$-DPP paths} have starting points $A_i=(0,i-2)$ and end points $E_i=(i,0)$ for $ i \in S$ where $S \subseteq \{2,\ldots,n\}$ and use steps in $(1,0)$-direction and $(0,-1)$-direction (note that our convention differs from the one more frequently used by the reflection along the line $y = x$.
%
%
\begin{figure}
\centering
\begin{tikzpicture}[xscale=-1]
\begin{scope}[scale=.55]
\TilingBox{1}{6}{4}
\begin{scope}[yshift=-1cm, xshift={-2*cos(30)*1cm}]\TilingBox{6}{4}{1}\end{scope}
\begin{scope}[yshift=-2cm]\TilingBox{4}{1}{6}\end{scope}
\PlanePartition{{4,4,4,2},{4,4,3,0},{4,3,3,-1},{4,3,3},{3,3,3,3,0},{3,2,1,1,0,0}}
\draw[fill=white] (0,0) -- ({-2*cos(30)},-1) -- (0,-2);
\RightBoundary{4}{-1}{0}
\RightBoundary{4}{-1}{-1}
\RightBoundary{4}{-2}{0}
\RightBoundary{4}{-2}{-1}
\TopBoundary{4}{-2}{0}
\RightBoundary{4}{-3}{-1}
\TopBoundary{4}{-3}{-1}
\LeftBoundary{4}{-3}{-1}
\LeftBoundary{3}{-3}{-1}
\LeftBoundary{3}{-3}{0}
\LeftBoundary{3}{-3}{1}
\draw[line width=1.5pt, red] (0,0) -- (0,4) -- ({-6*cos(30)},{4-6*sin(30)}) -- ({-6*cos(30)},{-6*sin(30)}) -- (0,0) -- (0,4);

\draw[line width=1.3pt, cyan] ({-.5*cos(30)},{3.5-.5*sin(30)}) -- ({-3.5*cos(30)},{3.5-3.5*sin(30)})  -- ({-3.5*cos(30)},{2.5-3.5*sin(30)})  -- ({-5.5*cos(30)},{2.5-5.5*sin(30)})  -- ({-5.5*cos(30)},{.5-5.5*sin(30)});
\draw[line width=1.3pt, cyan] ({-.5*cos(30)},{2.5-.5*sin(30)}) -- ({-.5*cos(30)},{1.5-.5*sin(30)}) -- ({-3.5*cos(30)},{1.5-3.5*sin(30)}) -- ({-3.5*cos(30)},{.5-3.5*sin(30)}) -- ({-4.5*cos(30)},{.5-4.5*sin(30)});
\draw[line width=1.3pt, cyan] ({-.5*cos(30)},{.5-.5*sin(30)}) -- ({-2.5*cos(30)},{.5-2.5*sin(30)}) ; 
\end{scope}
%
%
\begin{scope}[xshift=-4.5cm, yshift=-1.5cm, scale=.55]
\foreach \j in {0,...,3}{
	\draw[dashed] (-5.75,\j) -- (.75,\j);
}
\foreach \i in {-5,...,0}{
	\draw[dashed] (\i,-.75) -- (\i,3.75);
}
\draw[thick, ->] (0,0) -- (-6,0);
\draw[thick, ->] (0,0) -- (0,4);
\draw[line width=1.5pt, cyan] (0,3) -- (-3,3) -- (-3,2) -- (-5,2) -- (-5,0);
\draw[line width=1.5pt, cyan] (0,2) -- (0,1) -- (-3,1) -- (-3,0) -- (-4,0);
\draw[line width=1.5pt, cyan] (0,0) -- (-2,0);

\filldraw[black] (0,1) circle (3pt);
\filldraw[black] (-3,0) circle (3pt);
\filldraw[red] (0,0) circle (3pt);
\filldraw[red] (0,2) circle (3pt);
\filldraw[red] (0,3) circle (3pt);
\filldraw[red] (-2,0) circle (3pt);
\filldraw[red] (-4,0) circle (3pt);
\filldraw[red] (-5,0) circle (3pt);
\end{scope}
\end{tikzpicture}
%\end{center}
\caption{\label{fig: lozenge tiling and DPP} A cyclically symmetric lozenge tiling  of a hexagon with the fundamental domain framed in red (left) and the corresponding $5$-DPP paths.}
\end{figure}
%
%
In \cref{sec:vsasms}, we consider \defn{vertically symmetric alternating sign matrices} (VSASMs). 
%These are ASMs with an additional reflective symmetry along an axis. 
Due to the vertical symmetry, the central column of a VSASM has to exist and is of the form $(1,-1,1,\dots,-1,1)^\top$; hence, VSASMs only exist for an odd order.
Amongst the different objects that are known to be equinumerous to ASMs, cyclically symmetric lozenge tilings of a hexagon with a central triangular hole are particularly well-suited for imposing an additional condition that corresponds to the vertical symmetry of VSASMs. In fact, the axial symmetry can be literally transferred to lozenge tilings: it has been shown \cite{MRR87,Kup02} that $(2n+1) \times (2n+1)$ VSASMs are equinumerous with cyclically symmetric lozenge tilings of a hexagon with sides of alternating lengths $2n+2$ and $2n$ and with a central triangular hole of side length~$2$ that exhibit an additional reflective symmetry. \cref{fig:verticalSymmetricTiling} illustrates an example of such a tiling.
%
%
\begin{figure}[htb]
	\centering
		\begin{tikzpicture}[xscale=-1]
		\begin{scope}[scale=.55]
			\TilingBox{1}{6}{4}		
			\begin{scope}[yshift=-1cm, xshift={-2*cos(30)*1cm}]\TilingBox{6}{4}{1}\end{scope}
			\begin{scope}[yshift=-2cm]\TilingBox{4}{1}{6}\end{scope}
			\PlanePartition{{4,4,3,1},{4,3,1,0},{4,3},{3,2,2,2},{3,2,2,1,1},{2,1,1,0,0,0}}
			\draw[fill=white] (0,0) -- ({-2*cos(30)},-1) -- (0,-2);
			\draw[line width=1.3pt, cyan]({-1.5*cos(30)},{.5-1.5*sin(30)}) -- ({-2.5*cos(30)},{.5-2.5*sin(30)});
			\draw[line width=1.3pt, cyan]({-2.5*cos(30)},{2.5-2.5*sin(30)}) -- ({-4.5*cos(30)},{2.5-4.5*sin(30)}) -- ({-4.5*cos(30)},{1.5-4.5*sin(30)});
			
			\RightBoundary{4}{-1}{0}
			\RightBoundary{4}{-1}{-1}
			\RightBoundary{4}{-2}{-1}
			\LeftBoundary{4}{-2}{-1}
			\RightBoundary{3}{-2}{0}
			\LeftBoundary{3}{-2}{1}
			\LeftBoundary{3}{-2}{0}
			\draw[line width=1pt,dotted] ({-8*cos(30)},{-2*sin(30)}) -- ({6*cos(30)},{-2*sin(30)});
			\draw[line width=1.5pt, red] ({-cos(30)},{-sin(30)}) -- ({-3*cos(30)},{4-3*sin(30)}) -- ({-6*cos(30)},{4-6*sin(30)}) -- ({-6*cos(30)},{-2*sin(30)}) -- ({-2*cos(30)},{-2*sin(30)}) -- cycle;
	\end{scope}
	
	\begin{scope}[xshift=-5cm, yshift=-1.1cm, scale=.55]
\foreach \j in {0,...,2}{
	\draw[dashed] (-3.75,\j) -- (.75,\j);
}
\foreach \i in {-3,...,0}{
	\draw[dashed] (\i,-.75) -- (\i,2.75);
}
\draw[thick, ->] (0,0) -- (-4,0);
\draw[thick, ->] (0,0) -- (0,3);
\draw[line width=1.5pt, cyan] (0,0) -- (-1,0);
\draw[line width=1.5pt, cyan] (-1,2) -- (-3,2) -- (-3,1);

\filldraw[red] (0,0) circle (3pt);
\filldraw[red] (-1,2) circle (3pt);
\filldraw[red] (-1,0) circle (3pt);
\filldraw[red] (-3,1) circle (3pt);
\end{scope}
		\end{tikzpicture}
	%\end{center}
	\caption{Cyclically symmetric lozenge tiling with an additional reflective symmetry of a hexagon with alternating side lengths $2n+2$ and $2n$ for $n=2$. The fundamental domain framed in red corresponds to a sixth of the hexagon. The axis of reflection is indicated with a dotted line. The tiling corresponds to non-intersecting lattice paths from $(i-1,2i-2)$ to $(2i-1,i-1)$ for $1 \le i \le n$ with step set $\left\{ (1,0), (0,-1)\right\}$.}
	\label{fig:verticalSymmetricTiling}
\end{figure}
%
%
%
%\begin{figure}[htb]
%	\begin{center}
%		\begin{tikzpicture}[xscale=-1]
%		\begin{scope}[scale=.33]
%			\TilingBox{1}{8}{6}{3}			
%			\begin{scope}[yshift=-1cm, xshift={-2*cos(30)*1cm}]\TilingBox{8}{6}{1}{3}\end{scope}
%			\begin{scope}[yshift=-2cm]\TilingBox{6}{1}{8}{3}\end{scope}
%			\PlanePartitionWhite{{6,6,6,5,3,2},{6,6,5,2,2,1},{6,6,5,2,2,-1},{6,5,5},{5,3,3,3,3},{4,3,3,3,1,1,1,0},{4,3,3,3,1,0,0,0},{3,2,2,1,0,0,0,0}}
%			\draw[fill=white] (0,0) -- ({-2*cos(30)},-1) -- (0,-2);
%			\LeftBoundary{4}{-3}{1}{3}
%			\LeftBoundary{5}{-3}{0}{3}
%			\LeftBoundary{6}{-3}{-1}{3}
%			\LeftBoundary{5}{-3}{1}{3}
%			\LeftBoundary{4}{-3}{0}{3}
%			\LeftBoundary{4}{-3}{2}{3}
%			\RightBoundary{6}{-3}{-1}{3}
%			\RightBoundary{6}{-2}{-1}{3}
%			\RightBoundary{6}{-2}{0}{3}
%			\RightBoundary{6}{-1}{-1}{3}
%			\RightBoundary{6}{-1}{0}{3}
%			
%			\draw[line width=1.5pt,dotted] ({-10*cos(30)},{-2*sin(30)}) -- ({8*cos(30)},{-2*sin(30)});
%			\draw[line width=1.5pt, red] ({-cos(30)},{-sin(30)}) -- ({-4*cos(30)},{6-4*sin(30)}) -- ({-8*cos(30)},{6-8*sin(30)}) -- ({-8*cos(30)},{-2*sin(30)}) -- ({-2*cos(30)},{-2*sin(30)}) -- cycle;
%			\draw[line width=2pt, cyan] ({-1.5*cos(30)},{-.5*sin(30)}) -- ({-2.5*cos(30)},{.5-2.5*sin(30)});
%			\draw[line width=2pt, cyan] ({-2.5*cos(30)},{2.5*sin(30)}) -- ({-2.5*cos(30)},{.5*sin(30)}) -- ({-4.5*cos(30)},{.5-2.5*sin(30)});
%			\draw[line width=2pt, cyan] ({-3.5*cos(30)},{5.5*sin(30)}) -- ({-4.5*cos(30)},{2.5-.5*sin(30)}) -- ({-4.5*cos(30)},{2.5-2.5*sin(30)}) -- ({-6.5*cos(30)},{3-5.5*sin(30)}) -- ({-6.5*cos(30)},{3-7.5*sin(30)});
%			\end{scope}
%			\begin{scope}[scale=.6]
%		
%			%\draw[line width=1.5pt,dotted] ({4*cos(30)},{12*sin(30)}) -- ({-5*cos(30)},{-15*sin(30)});
%
%			
%			\node at (-8,-1) {$\Rightarrow$};
%			
%			\begin{scope}[xshift=-12cm, yshift=-3cm]
%				\foreach \j in {-0,...,4}{
%					\draw[dashed] (-5.75,\j) -- (.75,\j);
%				}
%				\foreach \i in {-5,...,0}{
%					\draw[dashed] (\i,-.75) -- (\i,4.75);
%				}
%				\draw[line width=2pt, cyan] (-2,4) -- (-3,4) -- (-3,3) -- (-5,3) -- (-5,2);
%				\draw[line width=2pt, cyan] (-1,2) -- (-1,1) -- (-3,1);
%				\draw[line width=2pt, cyan] (0,0) -- (-1,0);
%			\end{scope}
%			\end{scope}
%			
%		\end{tikzpicture}
%	\end{center}
%	\caption{alt}
%	\label{fig:verticalSymmetricTiling}
%\end{figure}