We have indicated in \cref{sec:mts} that there is a one-to-one correspondence between monotone triangles with bottom row $(1,2,\dots,n)$ and $n \times n$ ASMs. By applying similar ideas, we see that $(2n+1) \times (2n+1)$ VSASMs are in bijection with monotone triangles with bottom row $(0,2,\dots,2n-2)$: First we rotate the given matrix by $90^\circ$ and then map it to a monotone triangle with $2n+1$ rows of which we only need to consider the top $n$ rows due to the axial symmetry of the original VSASM. Since the central column of the VSASM was $(1,-1,1,-1,\dots,-1,1)^\top$, the top $n$ rows constitute a monotone triangle with bottom row $(2,4,\dots,2n)$. Finally, we subtract $2$ from each entry. 
%For instance, the subsequent VSASMs corresponds to the monotone triangle thus underlying the AMT on the right side of \cref{fig:amt}.

%\[
%\begin{pmatrix}
%	0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
%	0 & 1 & 0 & 0 &-1 & 0 & 0 & 1 & 0\\
%	0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
%	1 &-1 & 0 & 1 &-1 & 1 & 0 &-1 & 1\\
%	0 & 0 & 1 &-1 & 1 &-1 & 1 & 0 & 0\\
%	0 & 1 & 0 & 0 &-1 & 0 & 0 & 1 & 0\\
%	0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
%	0 & 0 & 0 & 1 &-1 & 1 & 0 & 0 & 0\\
%	0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0
%\end{pmatrix} 
%\quad \Rightarrow \quad
%\begin{array}{ccc ccc c}
%	&&&         4\\
%	&&       2 && 6 \\
%	&     2 && 3 && 6\\
%	0 && 2 && 4 && 6 
%\end{array}
%\]

Thanks to the correspondence between VSASMs and these monotone triangles, \cref{thm: amt gen fct} yields an enumerational refinement of $(2n+1)\times(2n+1)$ VSASMs by $n+3$ parameters. An obvious question is whether there is a corresponding family of non-intersecting lattice paths --- that might be interpreted as some kind of plane partitions or lozenge tilings afterwards --- with equally distributed statistics. There are indeed several such objects but just like we needed to broaden the notion of monotone triangles in favour of AMTs in order to incorporate additional statistics, we need to consider here more intricate families of non-intersecting lattice paths than those we get from lozenge tilings; see \cref{fig:verticalSymmetricTiling}. However, the weights of these lattice paths are signed. In \cref{thm:VSLatticePaths}, we provide one such family of non-intersecting lattice paths, an example of which is depicted in \cref{fig:ExampleVSLatticePaths}. By eliminating the sign, these paths can then be further transformed into pairs of certain plane partitions in \cref{thm:PairsPlanePartitions}.

\begin{thm}[{\cite[Theorem 2.2]{FH22}}]
	\label{thm:VSLatticePaths} 
	The generating function of AMTs with bottom row 
	$(0, 2 , \dots , 2n\linebreak-2)$ is equal to the signed generating function of $n$ lattice paths with starting points 
	$(-1,1)$, $(-2,2)$,\dots,$(-n,n)$ and end points $(0,1)$, $(1,0)$,\dots,$(n-1,-n+2)$ such that:
	\begin{itemize}
	\item[(i)] In the region $\{(x,y)|x \le 0\}$, the step set is $\{(1,1),(1,0)\}$, and steps of type $(1,0)$ are equipped with the weight~$w$.
	\item[(ii)] In the region $\{(x,y)| x \ge 0, y \ge 1\}$, the step set is $\{(1,-1),(0,-2)\}$, and steps of type $(0,-2)$  are equipped the weight $-uv$. 
        \item[(iii)] In the region $\{(x,y)| x \ge 0, y \le 1\}$, the step set is $\{(-1,0)^2,(0,-1)\}$, and one type of horizontal steps with distance $d$ from the line $y=2$ is equipped with the weight $u x_d$ and the second type is equipped with the weight $v x_d^{-1}$. 
        \end{itemize}  
	The paths are non-intersecting in the first and in the third region. In the second region, we distinguish between even and odd paths depending on whether they contain only even or only odd lattice points, respectively. Lattice paths of the same type are not intersecting each other, but an odd path may have an intersection with an even path. 
	
	The weight of a family of lattice paths is $\prod_{i=1}^{n} x_i^{n-1}$ multiplied by the product of the weights of all its steps where the weight of a step is $1$ if it has not been specified.
	Let $\sigma$ be the permutation so that the $i$-th starting point is connected to the $\sigma(i)$-th end point, then the sign of the family is $\sgn \sigma$. 
\end{thm}
%
\begin{figure}[htb]
	\centering
	\begin{tikzpicture}[scale=.5,baseline=(current bounding box.center)]
		\fill [verylight-gray] (-5.65,-3.5) rectangle (0,9.65);
		\fill [light-gray] (0,-3.5) rectangle (6.65,1);
		
		%\draw [help lines,step=1cm,dashed] (-5.75,-3.75) grid (6.75,9.75);
		\foreach \j in {-3,...,9}{
			\draw[dashed, gray!90!cyan] (-5.65,\j) -- (6.65,\j);
		}
		\foreach \i in {-5,...,6}{
			\draw[dashed, gray!90!cyan] (\i,-3.5) -- (\i,9.65);
		}
		
		\draw[->,thick] (-5.65,0)--(6.65,0);
		\draw[->,thick] (0,-3.5)--(0,9.65);
		
		\fill (-1,1) circle (5pt);
		\fill (-2,2) circle (5pt);
		\fill (-3,3) circle (5pt);
		\fill (-4,4) circle (5pt);
		\fill (-5,5) circle (5pt);
		
		\fill (0,1) circle (5pt);
		\fill (1,0) circle (5pt);
		\fill (2,-1) circle (5pt);
		\fill (3,-2) circle (5pt);
		\fill (4,-3) circle (5pt);
		
		\path[decoration=arrows, decorate] (-1,1) --++ (1,0);
		\draw[line width=1.3pt] (-1,1) --++ (1,0);
		
		\path[decoration=arrows, decorate] (-2,2) --++ (1,1) --++ (1,1);
		\draw[line width=1.3pt] (-2,2) --++ (1,1) --++ (1,1);
		\path[decoration=arrowsEven, decorate] (0,4) --++ (1,-1) --++ (1,-1) --++ (1,-1);
		\draw[line width=1.3pt,cyan] (0,4) --++ (1,-1) --++ (1,-1) --++ (1,-1);
		\path[decoration=arrows, decorate] (3,1) --++ (0,-1) --++ (-1,0) --++ (0,-1);
		\draw[line width=1.3pt] (3,1) --++ (0,-1) --++ (-1,0) --++ (0,-1);
		
		\path[decoration=arrows, decorate] (-3,3) --++ (1,0) --++ (1,1) --++ (1,1);
		\draw[line width=1.3pt] (-3,3) --++ (1,0) --++ (1,1) --++ (1,1);
		\path[decoration=arrowsOdd, decorate] (0,5) --++ (1,-1) --++ (0,-2) --++ (1,-1);
		\draw[line width=1.3pt,magenta] (0,5) --++ (1,-1) --++ (0,-2) --++ (1,-1);
		\path[decoration=arrows, decorate] (2,1) --++ (-1,0) --++ (0,-1);
		\draw[line width=1.3pt] (2,1) --++ (-1,0) --++ (0,-1);
		
		\path[decoration=arrows, decorate] (-4,4) --++ (1,1) --++ (1,0) --++ (1,0) --++ (1,1);
		\draw[line width=1.3pt] (-4,4) --++ (1,1) --++ (1,0) --++ (1,0) --++ (1,1);
		\path[decoration=arrowsEven, decorate] (0,6) --++ (1,-1) --++ (1,-1) --++ (1,-1) --++ (1,-1) --++ (1,-1);
		\draw[line width=1.3pt,cyan] (0,6) --++ (1,-1) --++ (1,-1) --++ (1,-1) --++ (1,-1) --++ (1,-1);
		\path[decoration=arrows, decorate] (5,1) --++ (0,-1) --++ (0,-1) --++ (0,-1) --++ (-1,0) --++ (0,-1);
		\draw[line width=1.3pt] (5,1) --++ (0,-1) --++ (0,-1) --++ (0,-1) --++ (-1,0) --++ (0,-1);
		
		\path[decoration=arrows, decorate] (-5,5) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,0) --++ (1,1);
		\draw[line width=1.3pt] (-5,5) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,0) --++ (1,1);
		\path[decoration=arrowsOdd, decorate] (0,9) --++ (1,-1) --++ (0,-2) --++ (1,-1) --++ (1,-1) --++ (0,-2) --++ (1,-1);
		\draw[line width=1.3pt,magenta] (0,9) --++ (1,-1) --++ (0,-2) --++ (1,-1) --++ (1,-1) --++ (0,-2) --++ (1,-1);
		\path[decoration=arrows, decorate] (4,1) --++ (0,-1) --++ (0,-1) --++ (-1,0) --++ (0,-1);
		\draw[line width=1.3pt] (4,1) --++ (0,-1) --++ (0,-1) --++ (-1,0) --++ (0,-1);	
	\end{tikzpicture}
	\caption{A family of non-intersecting lattice paths as defined in \cref{thm:VSLatticePaths} for $n=5$. In the second region, we draw even paths in red and odd paths in blue. The associated permutation is $\sigma = (1\;3\;2\;5\;4)$, and the weight is $(-u v)^3 u^4 w^4 x_1^5 x_2^5 x_3^5 x_4^5 x_5^4$ when always choosing the first type for horizontal steps in the third region.}
	\label{fig:ExampleVSLatticePaths}
\end{figure}
%
In Section~5 of \cite{FH22}, we have related the lattice paths of \cref{thm:VSLatticePaths} to non-intersecting lattice paths from $\left\{ (i-1,2i-2) \vert 1 \le i \le n\right\}$ to $\left\{ (2i-1,i-1) \vert 1 \le i \le n\right\}$ with step set $\left\{ (1,0), (0,-1)\right\}$ by purely combinatorial means. The latter are in one-to-one correspondence with cyclically and vertically symmetric lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size~$2$ as described in \cref{fig:verticalSymmetricTiling}. For this purpose, we have to consider the unrefined case by setting $u=v=1$, $w=-1$ and $x_1=\dots=x_n=1$; compare with \eqref{eq:explicit weights}.



In order to present a class of objects with signless weights, we recall the notion of column-strictness and row-strictness regarding plane partitions. A column-strict plane partition is a filling of a Young diagram with positive integers that weakly decrease along rows and strictly decrease down columns, whereas a row-strict plane partition is a filling of a Young diagram with positive integers that weakly decrease down columns and strictly along rows. The following described pairs of plane partitions provide a signless generating function that coincides with the $(n+3)$-parameter refinement of VSASMs. \cref{fig:PairsPlanePartitions} shows an example of such a pair.

\begin{thm}[{\cite[Theorem 2.5]{FH22}}] 
	\label{thm:PairsPlanePartitions} 
	The generating function of AMTs with bottom row 
	$(0,2,\dots,\allowbreak 2n-2)$ is equal to the generating function of pairs $(P,Q)$ of plane partitions of the same shape with $n$ rows (allowing also rows of length zero) such that $P$ is a column-strict plane partition and $Q$ is a row-strict plane partition, and the entries of $P$ in the $i$-th row from the bottom are no greater than $2i$, while the entries of $Q$ in the $i$-th row from the bottom are no greater than $i$. The weight of such a pair is given by the following monomial:
	$$
	w^{\binom{n+1}{2}-\# \text{ entries in $Q$}} \prod_{i=1}^{n} x_i^{n-1} (u x_i)^{\# \text{ ($2i-1$) in $P$}} (v x_i^{-1})^{\# \text{ ($2i$) in $P$}}.
	$$
\end{thm}
%
\begin{figure}[htb] 
	\centering \footnotesize
	\[
	\begin{ytableau}
		9 & 7 & 7 & 3 \\
		8 & 6 \\
		5 & 5 \\
		2
	\end{ytableau} 
	\qquad
	\begin{ytableau}
		4 & 3 & 2 & 1 \\
		4  & 2 \\
		3  & 2 \\
		1
	\end{ytableau} 
	\]
	\caption{Pair of a column-strict plane partition (left) and a row-strict plane partition (right) of the same shape for $n=5$. The weight of this pair is given by $w^{6} x_1^4 x_2^4 x_3^4 x_4^4 x_5^4 (v x_1^{-1})^1 (u x_2)^1 (u x_3)^2 (v x_3^{-1})^1 (u x_4)^2 (v x_4^{-1})^1 (u x_5)^1 $.}
	\label{fig:PairsPlanePartitions}
\end{figure}
%
We are able to prove \cref{thm:PairsPlanePartitions} in two different ways: one proof is rather based on algebraic manipulations, whilst the other one is more combinatorial. In fact, we can apply two sign-reversing involutions on the lattice paths of \cref{thm:VSLatticePaths} to obtain the pairs of plane partitions in \cref{thm:PairsPlanePartitions}. We essentially start with the family of lattice paths in \cref{thm:VSLatticePaths}, but where paths might intersect in the second and third region. The first sign-reversing involution eliminates intersections in the second region and shows that the generating function has no negative coefficients although each family of lattice paths is equipped with a sign. The intersections in the third regions are then dealt with by a second sign-reversing involution of ``Gessel-Viennot'' type.


In addition to providing a signless interpretation in terms of plane partitions, there is another benefit of \cref{thm:PairsPlanePartitions}: It yields an expansion of the generating function of AMTs with bottom row $(0,2,\dots,2n-2)$ into symplectic characters since column-strict plane partitions as they appear in \cref{thm:PairsPlanePartitions} are in bijective correspondence with symplectic tableaux as described by Koike and Terada \cite{KT90}. The coefficients in this expansion are given by \defn{totally symmetric self-complementary plane partitions}. This can be seen as follows: Given a row-strict plane partition with $n$ rows (including rows of length zero) and entries in the $i$-th row from the bottom not being greater than $i$, we replace each entry $\pi$ by $n+1-\pi$ and conjugate the resulting array such that we obtain a semistandard Young tableau with entries from $1$ to $n$ such that the entries in the $i$-th column are no smaller than $i$. By applying the standard procedure of mapping semistandard Young tableaux into Gelfand-Tsetlin patterns and adding an additional diagonal and adding $1$ to each entry, we obtain so-called \defn{Magog triangles} --- triangular arrays of positive integers with weakly increasing $\nearrow$- and $\searrow$-diagonals such that no entry in the $i$th $\nearrow$-diagonal from the left exceeds $i$ for every $i$ ---  which are known to be in bijection with totally symmetric self-complementary plane partitions in a $(2n+2) \times (2n+2) \times (2n+2)$ box \cite{MilRobRum86}. We give the Magog triangle that corresponds to the row-strict plane partition in \cref{fig:PairsPlanePartitions} below. Note that a similar result is known for the ordinary case: the generating function of AMTs with bottom row $(1,2,\dots,n)$ exhibits an expansion into Schur functions whose coefficients are given by weights of \defn{totally symmetric plane partitions} \cite{TSPPmany}.


%In addition to providing a signless interpretation in terms of plane partitions, there is another benefit of \cref{thm:PairsPlanePartitions}: It yields an expansion of the generating function of AMTs with bottom row $(0,2,\dots,2n-2)$ into symplectic characters since column-strict plane partitions as they appear in \cref{thm:PairsPlanePartitions} are in bijective correspondence with symplectic tableaux as described by Koike and Terada \cite{KT90}. The coefficients in this expansion are given by \defn{totally symmetric self-complementary plane partitions}. This can be seen as follows: Given a row-strict plane partition with $n$ rows (including rows of length zero) and entries in the $i$-th row from the bottom not being greater than $i$, we replace each entry $\pi$ by $n+2-\pi$ and conjugate the resulting array such that we obtain a semistandard Young tableau with entries from $2$ to $n+1$ such that the entries in the $i$-th column are no smaller than $i+1$. By applying the standard procedure of mapping semistandard Young tableaux into Gelfand-Tsetlin patterns and adding $1$ to each entry, we obtain so-called \defn{Magog triangles} --- triangular arrays of positive integers with weakly increasing $\nearrow$- and $\searrow$-diagonals such that no entry in the $i$th $\nearrow$-diagonal from the left exceeds $i$ for every $i$ ---  which are known to be in bijection with totally symmetric self-complementary plane partitions in a $(2n+2) \times (2n+2) \times (2n+2)$ box \cite{MilRobRum86}. We give the Magog triangle that corresponds to the row-strict plane partition in \cref{fig:PairsPlanePartitions} below. Note that a similar result is known for the ordinary case: the generating function of AMTs with bottom row $(1,2,\dots,n)$ exhibits an expansion into Schur functions whose coefficients are given by weights of \defn{totally symmetric plane partitions} \cite{TSPPmany}.

\[ \footnotesize
\begin{ytableau}
	4 & 3 & 2 & 1 \\
	4  & 2 \\
	3  & 2 \\
	1
\end{ytableau}
\quad \Rightarrow \quad
\begin{array}{ccc ccc ccc cc}
	&&&&&         1\\
	&&&&       1 && 1 \\
	&&&     1 && 1 && 3\\
	&&    1 && 1 && 2 && 4 \\
	&  1 && 1 && 2 && 4 && 4\\
	1 && 1 && 2 && 2 && 4 && 5
\end{array}
\]

