

We present the modified objects replacing %cyclically symmetric lozenge tilings of a hexagon with a triangular hole of side length $2$ 
DPPs along with the multivariate weight with $n+3$ variables that give rise to the $n+3$ statistics on the DPP-side.
 
\defn{Extended $n$-DPP paths}  are families of non-intersecting lattice paths with starting points $A_i^\prime=(-i,i-2)$ and end points $E_i^\prime=(i,-2)$ for $i\in S$ and a given subset $S \subseteq \{1,\ldots,n\}$ which stay weakly below the line $y=n-2$ and have the following step sets and weights:
\begin{enumerate}
	\item[(i)] In the region $\{(x,y) |  x<0 \}$, the step set is $\{(1,0),(0,1)^2\}$ with weights $x_{p+q+2} v^{-1}$ for a $(1,0)$ step ending in $(p,q)$, and where $(1,0)^2$ indicates that there are two types of $(0,1)$ steps, one of which has weight 
	$x_{p+q+2} v^{-1} w$ when ending in $(p,q)$ and the other has weight $1$.
	\item[(ii)]  In the region $\{(x,y) |  x\geq 0, y \geq -1 \}$, the step set is $\{(1,0),(0,-1)\}$, with weights $u x_{q+2}$ for a $(1,0)$ step at height $q$ and weight $1$ for a $(0,-1)$ step.	
	\item[(iii)]  If we go below the line $y=-1$, the step set is $\{(1,-1),(0,-1)\}$, with weights $w$ for a $(1,-1)$ step and $1$ for a $(0,-1)$ step.
\end{enumerate}
The weight of extended $n$-DPP paths is $v^{\binom{n+1}{2}}$ times the product of the weights of all steps. See \cref{fig:extended n-DPP path} for an example. Note that in \cite[Section 4]{nASMDPP} we consider a (very minor) modification of these paths. 
%
% We can consider a slight variation of extended DPP paths which results in monomial weights. 
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=.5]
\foreach \j in {-2,...,5}{
	\draw[dashed, gray!90!black] (-2.75-\j,\j) -- (7.75,\j);
}
\foreach \i in {0,...,7}{
	\draw[dashed, gray!90!black] (\i,-2.5) -- (\i,5.5);
}
\foreach \i in {-7,...,-1}{
	\draw[dashed, gray!90!black] (\i,-2.5-\i) -- (\i,5.5);
}


\draw[thick, ->] (-2,0) -- (8,0);
\draw[thick, ->] (0,-2) -- (0,5.75);

\draw[line width=1.5pt, cyan] (-1,-1) -- (0,-1) -- (1,-2);
\draw[line width=1.5pt, cyan] (-2,0) -- (-1,0) -- (-1,1) -- (1,1) -- (1,0) -- (2,0) -- (2,-2);
\draw[line width=1.5pt, cyan] (-4,2) -- (-2,2) -- (-2,3) -- (-1,3) -- (0,3) -- (0,2) -- (2,2) -- (2,1) -- (3,1) -- (3,-1) -- (4,-2);
\draw[line width=1.5pt, cyan] (-6,4) -- (-1,4) -- (-1,5) -- (2,5) -- (2,4) -- (3,4) -- (3,2) -- (4,2) -- (4,1) -- (5,1) -- (5,-1) -- (6,-1) -- (6,-2);

\filldraw[red] (1,-2) circle (4pt);
\filldraw[red] (-1,-1) circle (4pt);
\filldraw[red] (2,-2) circle (4pt);
\filldraw[red] (-2,0) circle (4pt);
\filldraw[red] (4,-2) circle (4pt);
\filldraw[red] (-4,2) circle (4pt);
\filldraw[red] (6,-2) circle (4pt);
\filldraw[red] (-6,4) circle (4pt);
\filldraw (3,-2) circle (4pt);
\filldraw (5,-2) circle (4pt);
\filldraw (7,-2) circle (4pt);
\filldraw (-3,1) circle (4pt);
\filldraw (-5,3) circle (4pt);
\filldraw (-7,5) circle (4pt);


%\foreach \j in {-2,...,3}{
%	\draw[dashed, gray!90!black] (-2.75-\j,\j) -- (5.75,\j);
%}
%\foreach \i in {0,...,5}{
%	\draw[dashed, gray!90!black] (\i,-2.5) -- (\i,3.5);
%}
%\foreach \i in {-5,...,-1}{
%	\draw[dashed, gray!90!black] (\i,-2.5-\i) -- (\i,3.5);
%}
%
%
%\draw[thick, ->] (-2,0) -- (6,0);
%\draw[thick, ->] (0,-2) -- (0,3.75);
%
%\draw[line width=1.5pt, cyan] (-1,-1) -- (0,-1) -- (1,-2);
%\draw[line width=1.5pt, cyan] (-2,0) -- (-1,0) -- (-1,1) -- (1,1) -- (1,0) -- (2,0) -- (2,-2);
%\draw[line width=1.5pt, cyan] (-4,2) -- (-2,2) -- (-2,3) -- (-1,3) -- (0,3) -- (0,2) -- (2,2) -- (2,1) -- (3,1) -- (3,-1) -- (4,-2);
%
%\filldraw[red] (1,-2) circle (4pt);
%\filldraw[red] (-1,-1) circle (4pt);
%\filldraw[red] (2,-2) circle (4pt);
%\filldraw[red] (-2,0) circle (4pt);
%\filldraw[red] (4,-2) circle (4pt);
%\filldraw[red] (-4,2) circle (4pt);
%\filldraw (3,-2) circle (4pt);
%\filldraw (5,-2) circle (4pt);
%\filldraw (-3,1) circle (4pt);
%\filldraw (-5,3) circle (4pt);


\end{tikzpicture}
%\end{center}
\caption{\label{fig:extended n-DPP path} A family of extended $5$-DPP paths with $S=\{1,2,4,6\}$ and weight $u^{11}v^5 w^5 x_1^5x_2^4 x_3^6 x_4^5x_5^2x_6^2 x_7^3$ where all $(0,1)$ steps left of the $y$-axis are of first type.}
\end{figure}
%1 2 4 5; 1 3; 1
%x_1
%
Analogously to the situation for AMTs and ASMs, we argue next that the generating function of extended $n$-DPP paths is the number of $n$-DPP paths under the specialization \eqref{eq:explicit weights}. Indeed, strictly left of the $y$-axis $(0,1)$ steps come in pairs with weights of opposite sign; hence, left of the $y$-axis, we can assume to have only $(1,0)$ steps. The following is a sign-reversing involution between all such extended $n$-DPP paths for which not all paths end by two $(0,-1)$ steps (and hence cancels these paths): pick the right-most path ending by either a diagonal $(1,-1)$ step or  by a 
$(1,0)$ step followed by an $(0,-1)$ step and replace in the second case the two steps by a diagonal step and vice-versa. By deleting all steps left of the $y$-axis and below the $x$-axis of the remaining extended DPP paths, we obtain the above defined DPP paths.

%The remaining families of paths are exactly DPP paths, where we extend each path by a $(0,-1)$ step followed by either another $(0,-1)$ step, a $(1,0)$ step and a $(-1,0)$ step or a $(1,-1)$ step. The weight of these possibilities is $1,1,-1$ respectively; the total contribution is therefore $1$ for which reason we can omit also these steps and finally obtain a $n$-DPP path.\\

Our second main result is a consequence of \cite[Theorem 2.6]{nASMDPP} using insights from Section~4 in that paper.

\begin{thm}
\label{thm:ASMs and DPPs}
The generating function of AMTs with bottom row $(1,2,\ldots,n)$ is equal to the generating function of extended $n$-DPP paths.
\end{thm}

It is an open problem to construct a weight-preserving bijection between AMTs with bottom row $(1,2,\ldots,n)$ and extended $n$-DPP paths.
One surprising observation we made in our work is the following: Suppose that one is able to find a weight preserving bijection between AMTs with bottom row $(1,2,\ldots,n)$ and extended $n$-DPP paths, which is further compatible with the sign reversing involution described above and the one hinted at in the paragraph around \eqref{eq:explicit weights}. Then, as a direct consequence, one would obtain a longed for bijection between ASMs and DPPs. However, we show that such a compatible weight preserving bijection does not exist for $n \ge 3$.


\begin{prop}
\label{prop: no bijection}
Let $n \geq 3$. Then there exists no weight-preserving bijection between AMTs with bottom row $(1,\ldots,n)$ and extended $n$-DPP paths which induces a bijection between monotone triangles with bottom row $(1,\ldots,n)$ and $n$-DPP paths by restricting to $n$-DPP paths as described above and by ignoring the arrows in the corresponding AMT.
\end{prop}
\begin{proof}
We extend $n$-DPP paths to extended $n$-DPP paths by adding an appropriate number of $(1,0)$ steps at the beginning of each path and by adding two $(0,-1)$ steps at the end of each path. The exponent of $x_1$ thereby counts the number of paths in a given family of $n$-DPP paths. Hence the maximum exponent of $x_1$ is $n-1$ and obtained by the unique family of $n$-DPP paths consisting of $n-1$ paths.
%\flo{}
For AMTs, the exponent of $x_1$ is $n-1$ exactly if the top entry is $n$. The number of monotone triangles with top entry $n$ is equal to the number of monotone triangles with bottom row $(1,\ldots,n-1)$ since the top entry $n$ forces all entries in the right-most northwest-diagonal to be equal to $n$. For $n \geq 3$, there are at least $2$ of such monotone triangles. Hence not all of them can be reached through a weight-preserving bijection.
\end{proof}

There exists also a version of \cref{thm:ASMs and DPPs} in terms of certain plane partitions, which we discuss next. 
Let $\lambda$ be a partition and $(a_1,\ldots,a_l|b_1,\ldots,b_l)$ its Frobenius notation, i.e., $a_i=\lambda_i-i$ and $b_i=\lambda^\prime_i-i$, where $l=\max_i\{\lambda_i \geq i\}$.
We say that $\lambda$ is \defn{near-balanced} if, for all $i$, either $a_i=b_i$ or $a_i=b_i+1$. We assign the weight 
$\W(\lambda) = w^{l+\sum_{i=1}^l (b_i - a_i)}$. Now, a \defn{set-valued near-balanced column strict plane partition} (SBCSPP) $D$ of shape $\lambda$ and order $n$ is a filling of a near-balanced  partition $\lambda$ with non-empty subsets of  $\{1,2,\ldots,n\}$ such that strictly above the diagonal the subsets are singletons, and (i) rows decrease weakly in the sense that the maxima of the sets form a weakly decreasing sequence if read from left to right, and (ii) columns decrease strictly in the sense that for two adjacent cells in a column, all elements in the top cell are strictly greater than all elements in the bottom cell. The weight of $D$ is as follows
\begin{multline*} 
\W(D) = \W(\lambda) \cdot u^{\# \text{ cells strictly above the main diagonal}}  
\cdot v^{\binom{n+1}{2} -\# \text{ entries on and below the main diagonal}} \\
\cdot w^{\# \text{ entries} - \# \text{ cells}} \cdot \prod_{i=1}^n x_i^{\# i \text{ in D}}.
\end{multline*} 
%An example of an SBCSPP is displayed next.
Next is an example of an SBCSPP of order $9$ and weight $u^{16} v^{26} w^{3} x_1^{5} x_2^{5} x_3^{4} x_4^5 x_5^4 x_6^4 x_7^5 x_8^3$.
$$
\footnotesize
\begin{ytableau}
 8 & 8 & 8 & 7 & 7 & 6 & 4 & 1 \\
 7 &  7  & 7 & 6 & 5 & 5  \\
  6 &   6 &  5  & 4 & 4 & 4  \\
 5 &  4 &  3 &  3,2  & 3 & 2 \\
 3 &  2 &  2,1 &  1 \\
 2 & 1 \\
 1
\end{ytableau}
$$
%Letting $n=9$, its weight is 
%$
%u^{16} v^{26} w^{3} x_1^{5} x_2^{5} x_3^{4} x_4^5 x_5^4 x_6^4 x_7^5 x_8^3.
%$

\begin{thm}[{\cite[Theorem 2.6]{nASMDPP}}]
\label{thm:ASMs and DPPs2}
The generating function of arrowed monotone triangles with bottom row $1,2,\ldots,n$ is equal to the generating function of set-valued near-balanced column strict plane partitions with parts in $\{1,2,\ldots,n\}$.
\end{thm} 

Concerning the proof of \cref{thm:ASMs and DPPs2} (and also of \cref{thm:VSLatticePaths}), the rough  idea is to use \cref{thm: amt gen fct} to obtain a bialternant formula for the generating function of AMTs. We then transform this generating function into a Jacobi-Trudi-type determinant, which we then interpret combinatorially using the Lindstr\"om-Gessel-Viennot theorem.





