Possible database queries for alternating sign matrices: search your data / browse all statistics / browse all maps

# 1. Definition & Example

• An alternating sign matrix (ASM) is a square matrix whose entries all belong to $\{-1,0,1\}$ such that the sum of each row and column is 1 and the non-zero entries in each row and column alternate in sign.

• The size of an alternating sign matrix is the size of the square matrix.

 The seven alternating sign matrices of size 3 [[1,0,0],[0,1,0],[0,0,1]] [[0,1,0],[1,0,0],[0,0,1]] [[1,0,0],[0,0,1],[0,1,0]] [[0,1,0],[1,-1,1],[0,1,0]] [[0,0,1],[1,0,0],[0,1,0]] [[0,1,0],[0,0,1],[1,0,0]] [[0,0,1],[0,1,0],[1,0,0]]
• The number of alternating sign matrices of size $n$ is $$\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!},$$ see A005130. This result was known as the alternating sign matrix conjecture before being proved in [Ze92].

# 2. FindStat representation and coverage

• An alternating sign matrix is uniquely represented as a list of lists representing its rows.

• Alternating sign matrices are graded by its size.

• The database contains all alternating sign matrices of size at most 6.

• The corner-sum matrix $(c_{i,j})_{i,j=0}^n$ of an ASM $(a_{i,j})_{i,j=1}^n$ of order $n$ is the $n+1 \times n+1$ matrix where $c_{i,j} = \sum_{i' \leq i, j' \leq j} a_{i',j'}$. For example, the ASM has corner-sum matrix

 $\left(\begin{smallmatrix} 0 &0 &1 \\1 &0 &0\\ 0 &1 &0\end{smallmatrix}\right)\rightarrow\left(\begin{smallmatrix} 0 &0 &0 &0 \\0 &0 &0 &1\\ 0 &1 &1 &2 \\ 0 &1 &2 &3 \end{smallmatrix}\right)$
• Given a corner-sum matrix $(c_{i,j})_{i,j=0}^n$ define $h_{i,j} = i+j-2c_{i,j}$. The matrix $(h_{i,j})_{i,j=0}^n$ is the height-function matrix. In reference to the above corner-sum matrix, we have the height-function matrix

 $\left(\begin{smallmatrix} 0 &0 &0 &0 \\0 &0 &0 &1\\ 0 &1 &1 &2 \\ 0 &1 &2 &3\end{smallmatrix}\right)\rightarrow\left(\begin{smallmatrix} 0 &1 &2 &3 \\1 &2 &3 &2\\ 2 &1 &2 &1 \\ 3 &2 &1 &0\end{smallmatrix}\right)$
• ASM's are in bijection with square-ice models (also known as the "6-vertex model") that satisfy the domain-wall boundary conditions.
• Define the matrix $A^n = (a_{i,j})_{i,j = 1}^n$ where $a_{i,j}$ is the number of $n \times n$ ASMs with a $1$ in the first row in column $i$ and a $1$ in the last row in column $j$. For example, $$A^1 = (1), A^2 = \begin{pmatrix} 0 &1 \\ 1 &0 \end{pmatrix}, A^3 = \begin{pmatrix} 0 &1 &1 \\ 1 &1 &1 \\ 1 &1 &0 \end{pmatrix}$$ We may also find the determinants of these matrices by using the following relation $\det(A^n) = (-A_{n-1})^{n-3}$.

[BCS].

# 4. Monotone triangles

Define $\left(s_{ij}\right)_{i,j=1, \dots, n}$ to be the $n \times n$ matrix whose entries are the partial sums of the columns (top to bottom) of an $n \times n$ alternating sign matrix. The monotone triangle of an alternating sign matrix is a triangular array whose $i^{\text{th}}$ row consists of the values $j$ in which the entry $s_{ij}=1$.

 $\left(\begin{smallmatrix} 0& 0& 0 &1 &0 \\ 0& 1& 0 &-1 &1 \\ 1& -1& 0 &1 &0 \\ 0& 0& 1 &0 &0 \\ 0& 1& 0 &1 &0 \end{smallmatrix}\right)\quad\quad\rightarrow\begin{smallmatrix}&&&&&&&&4&&&&&&&&\\&&&&&&&2&&5&&&&&&&\\&&&&&&1&&4&&5&&&&&&\\&&&&&1&&3&&4&&5&&&&&\\&&&&1&&2&&3&&4&&5&&&&\end{smallmatrix}$

# 5. Dyck Path Tuples

Applying Mp00007 to each row in an alternating sign matrix, you can construct a tuple of nested dyck paths.

# 6. References

[BCS]   Phillip Biane, Luigi Cantini, and Andrea Sportiello, Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions, http://arxiv.org/pdf/1101.3427v1.pdf .

[Ze92]   D. Zeilberger, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13.