A \defn{monotone triangle} is a triangular array $(m_{i,j})_{1 \leq j \leq i \leq n}$ of integers of the form 
%\[
%\begin{array}{ccccccccccccccccc}
%  &   &   &   &   &   &   &   & m_{1,1} &   &   &   &   &   &   &   & \\
%  &   &   &   &   &   &   & m_{2,1} &   & m_{2,2} &   &   &   &   &   &   & \\
%  &   &   &   &   &   & \dots &   & \dots &   & \dots &   &   &   &   &   & \\
%  &   &   &   &   & m_{n-2,1} &   & \dots &   & \dots &   & m_{n-2,n-2} &   &   &   &   & \\
%  &   &   &   & m_{n-1,1} &   & m_{n-1,2} &  &   \dots &   & \dots   &  & m_{n-1,n-1}  &   &   &   & \\
%  &   &   & m_{n,1} &   & m_{n,2} &   & m_{n,3} &   & \dots &   & \dots &   & m_{n,n} &   &   &
%\end{array}
%\]
\begin{center}
\begin{tikzpicture}[scale=0.6]
\node at (0,0) {$m_{n,1}$};
\node at (2,0) {$m_{n,2}$};
\node at (8,0) {$m_{n,n}$};
\node at (1,1) {$m_{n-1,1}$};
\node at (7,1) {$m_{n-1,n-1}$};
\draw[dashed, thick] (1.5,1.5) -- (2.5,2.5);
\draw[dashed, thick] (3,0) -- (7,0);
\draw[dashed, thick] (2.25,1) -- (5.5,1);
\draw[dashed, thick] (5.5,2.5) -- (6.5,1.5);
\node at (3,3) {$m_{2,1}$};
\node at (5,3) {$m_{2,2}$};
\node at (4,4) {$m_{1,1}$};
\end{tikzpicture}
\end{center}
with weakly increasing entries along northeast- and southeast-diagonals, i.e., $m_{i+1,j} \leq m_{i,j} \leq m_{i+1,j+1}$, and strictly increasing rows, i.e., $m_{i,j} < m_{i,j+1}$. 
It is well known that $n\times n$ ASMs are in bijection with monotone triangles with bottom row $(1,2,\ldots,n)$.
%by the following construction. In the first step replace each entry of the ASM by the sum of all entries weakly above in the same column; all entries are either $0$ or $1$ then. Now record for each row the columns with a $1$ entry. It is immediate that the resulting array is a monotone triangle and that the map is bijective. An example is given next.
%\[
%\begin{pmatrix}
%0 & 0 & 1 & 0 & 0 & 0\\
%0 & 1 &-1 & 0 & 1 & 0\\
%0 & 0 & 1 & 0 & 0 & 0\\
%1 &-1 & 0 & 1 & -1 & 1\\
%0 & 1 & 0 & 0 & 0 & 0\\
%0 & 0 & 0 & 0 & 1 & 0
%\end{pmatrix} 
%\; \Rightarrow \; 
%\begin{pmatrix}
%0 & 0 & 1 & 0 & 0 & 0\\
%0 & 1 & 0 & 0 & 1 & 0\\
%0 & 1 & 1 & 0 & 1 & 0\\
%1 & 0 & 1 & 1 & 0 & 1\\
%1 & 1 & 1 & 1 & 0 & 1\\
%1 & 1 & 1 & 1 & 1 & 1
%\end{pmatrix}
%\; \Rightarrow
%\begin{array}{ccc ccc ccc cc}
%&&&&&         3\\
%&&&&       2 && 5 \\
%&&&     2 && 3 && 5\\
%&&    1 && 3 && 4 && 6 \\
%&  1 && 2 && 3 && 4 && 6\\
%1 && 2 && 3 && 4 && 5 && 6
%\end{array}
%\]
%
%
As we see in \cref{sec:vsasms}, monotone triangles with bottom %row $(1,2,\ldots,n)$ are in bijection to $n \times n$ ASMs and monotone triangles with bottom  
row $(0,2,\ldots,2n-2)$ are in bijection with $(2n+1)\times(2n+1)$ VSASMs in a similar way. Our modified ASM-objects are the following decorated monotone triangles.



An \defn{arrowed monotone triangle} (AMT) is a monotone triangle $(m_{i,j})_{1 \leq j \leq i\leq n}$ together with a decoration of its entries by one of the symbols $\nwarrow, \nearrow, \nenwarrow$  such that the decoration of $m_{i+1,j}$ has to be $\nwarrow$ if $m_{i+1,j} = m_{i,j}$, while 
the decoration of $m_{i+1,j+1}$ has to be $\nearrow$ if $m_{i+1,j+1}= m_{i,j}$. In other words, an arrow indicates that the decorated entry has to be different from the entry it is pointing to.
%
%
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.55]
\node at (0,0) {$1$};
\node at (2,0) {$2$};
\node at (4,0) {$3$};
\node at (6,0) {$4$};
\node at (8,0) {$5$};
\node at (10,0) {$6$};
\node at (1,1) {$1$};
\node at (3,1) {$2$};
\node at (5,1) {$3$};
\node at (7,1) {$4$};
\node at (9,1) {$6$};
\node at (2,2) {$1$};
\node at (4,2) {$3$};
\node at (6,2) {$4$};
\node at (8,2) {$6$};
\node at (3,3) {$2$};
\node at (5,3) {$3$};
\node at (7,3) {$5$};
\node at (4,4) {$2$};
\node at (6,4) {$5$};
\node at (5,5) {$3$};

\node at (-.5,.45) {\footnotesize $\nwarrow$};
\node at (1.5,.45) {\footnotesize $\nwarrow$};
\node at (3.5,.45) {\footnotesize $\nwarrow$};
\node at (5.5,.45) {\footnotesize $\nwarrow$};
\node at (7.5,.45) {\footnotesize $\nwarrow$};
\node at (8.5,.45) {\footnotesize $\nearrow$};
\node at (10.5,.45) {\footnotesize $\nearrow$};

\node at (.5,1.45) {\footnotesize $\nwarrow$};
\node at (2.5,1.45) {\footnotesize $\nwarrow$};
\node at (5.5,1.45) {\footnotesize $\nearrow$};
\node at (7.5,1.45) {\footnotesize $\nearrow$};
\node at (9.5,1.45) {\footnotesize $\nearrow$};

\node at (1.5,2.45) {\footnotesize $\nwarrow$};
\node at (2.5,2.45) {\footnotesize $\nearrow$};
\node at (3.5,2.45) {\footnotesize $\nwarrow$};
\node at (6.5,2.45) {\footnotesize $\nearrow$};
\node at (7.5,2.45) {\footnotesize $\nwarrow$};

\node at (2.5,3.45) {\footnotesize $\nwarrow$};
\node at (4.5,3.45) {\footnotesize $\nwarrow$};
\node at (7.5,3.45) {\footnotesize $\nearrow$};

\node at (4.5,4.45) {\footnotesize $\nearrow$};
\node at (5.5,4.45) {\footnotesize $\nwarrow$};
\node at (6.5,4.45) {\footnotesize $\nearrow$};

\node at (4.5,5.45) {\footnotesize $\nwarrow$};

\begin{scope}[xshift=15cm]
\node at (0,0) {$0$};
\node at (2,0) {$2$};
\node at (4,0) {$4$};
\node at (6,0) {$6$};
\node at (1,1) {$2$};
\node at (3,1) {$3$};
\node at (5,1) {$6$};
\node at (2,2) {$2$};
\node at (4,2) {$6$};
\node at (3,3) {$4$};

\node at (.5,.45) {\footnotesize $\nearrow$};
\node at (2.5,.45) {\footnotesize $\nearrow$};
\node at (3.5,.45) {\footnotesize $\nwarrow$};
\node at (4.5,.45) {\footnotesize $\nearrow$};
\node at (6.5,.45) {\footnotesize $\nearrow$};

\node at (.5,1.45) {\footnotesize $\nwarrow$};
\node at (2.5,1.45) {\footnotesize $\nwarrow$};
\node at (5.5,1.45) {\footnotesize $\nearrow$};

\node at (2.5,2.45) {\footnotesize $\nearrow$};
\node at (3.5,2.45) {\footnotesize $\nwarrow$};

\node at (2.5,3.45) {\footnotesize $\nwarrow$};
\node at (3.5,3.45) {\footnotesize $\nearrow$};

\end{scope}
\end{tikzpicture}
%\end{center}
\caption{\label{fig:amt}An arrowed monotone triangle with bottom row $(1,2,3,4,5,6)$ on the left and with bottom row $(0,2,4,6)$ on the right.}
\end{figure}
%
%
We write ${^{\nwarrow}{m_{i,j}}, {m_{i,j}}^{\nearrow}}, {^{\nwarrow}{m_{i,j}}^{\nearrow}}$ if the entry $m_{i,j}$ is decorated with $\nwarrow, \nearrow$, or $\nenwarrow$, respectively. See \cref{fig:amt} for an example.
We assign to an AMT $M=(m_{i,j})_{1 \leq j \leq i \leq n}$ the weight
\[
W_M(u,v,w;\x)=  u^{\# \nearrow} v^{\# \nwarrow} w^{\# \nenwarrow}  \prod_{i=1}^n x_i^{\sum_{j=1}^i m_{i,j} - \sum_{j=1}^{i-1} m_{i-1,j} + \#(\nearrow \text{ in row } i) - \#(\nwarrow \text{ in row } i)}.
\]
The AMT in \cref{fig:amt} (left) has weight $u^7 v^{11} w^3 x_1^2 x_2^5 x_3^2 x_4^3 x_5^3 x_6^2$ and the AMT in  \cref{fig:amt} (right) has weight $u^5 v^3 w^2 x_1^4 x_2^4 x_3^2 x_4^4$.
%
The exponents of the $n+3$ variables are the $n+3$ statistics on the ASM-side of our story. Note that there is a very satisfying analogy to the weight of Gelfand-Tsetlin patterns that corresponds to the Schur weight on semistandard Young tableaux under the classical bijection as this weight is simply $\prod_{i=1}^n x_i^{\sum_{j=1}^i a_{i,j} - \sum_{j=1}^{i-1} a_{i-1,j}}$ for a given Gelfand-Tsetlin pattern $A=(a_{i,j})_{1 \leq j \leq i \leq n}$. 


AMTs and monotone triangles are related as follows. Fix a monotone triangle $M$ and consider the set of all AMTs which are mapped to $M$ by forgetting the decorations. We claim that the sum of weights of these AMTs is equal to $1$ when setting 
\begin{equation}
\label{eq:explicit weights}
u=v=1, \quad w=-1, \quad x_1=\ldots=x_n=1.
\end{equation}
Indeed, this follows from the observation that each entry of $M$ that is different from its northwest-neighbor and its northeast-neighbor can be decorated with any of
$\nwarrow, \nearrow, \nenwarrow$, while all the other entries can be decorated with either $\nwarrow$ or $\nearrow$.
(In \cite[Proposition 3.3]{nASMDPP} we also identify a weight on monotone triangles directly that is up to a simple factor the sum of weights of all arrowed monotone triangles that are obtained by decorating the monotone triangle.)
Therefore, we may consider AMTs with bottom row $(1,2,\ldots,n)$ as our modified objects on the ASM-side.

\bigskip

For a sequence $L=(L_1,\ldots,L_n)$ of integers (not necessarily decreasing), we define the \defn{extended Schur polynomial} $s_L(\x)$ as
$
s_L(\x) = \frac{\det\limits_{1 \leq i,j \leq n}\left(x_i^{L_j+n-j}\right)}{\prod_{1 \leq i < j \leq n}(x_i-x_j)}.
$
For an equivalent definition see \cite[Equation (3.6), (5.1)]{nASMDPP}. We are now able to state our first main result.

\begin{thm}[{\cite[Theorem 2.2]{nASMDPP}}]
\label{thm: amt gen fct}
The generating function of arrowed monotone triangles with bottom row $k_1 < \ldots < k_n$ is 
\begin{equation} 
\label{eq: amt gen fct}
\prod_{i=1}^{n} (u X_i + v X_i^{-1} + w) \\ 
\prod_{1 \le p < q \le n} \left( u  \e_{k_p} + v \e_{k_q}^{-1} + w \e_{k_p} \e_{k_q}^{-1}  \right) \\
s_{(k_n,k_{n-1},\ldots,k_1)}(x_1,\ldots,x_n), 
\end{equation} 
where $\e_k$ denotes the shift operator, defined as $\e_k p(k) = p(k+1)$. 
\end{thm} 

This is a multivariate extension of the operator formula of number of monotone triangles with bottom row \cite{Fis06} and one of the main ingredient in the proofs of our other results.

