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1. Definition & Example

The seven alternating sign matrices of size 3

http://www.findstat.org/Graphics/api/Cc0017_0_[[1,0,0],[0,1,0],[0,0,1]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[0,1,0],[1,0,0],[0,0,1]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[1,0,0],[0,0,1],[0,1,0]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[0,1,0],[1,-1,1],[0,1,0]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[0,0,1],[1,0,0],[0,1,0]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[0,1,0],[0,0,1],[1,0,0]].png

http://www.findstat.org/Graphics/api/Cc0017_0_[[0,0,1],[0,1,0],[1,0,0]].png

 [[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]] 

2. FindStat representation and coverage

3. Additional information

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

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

[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.

DyckPathTuples.png

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.

7. Sage examples


CategoryCombinatorialCollection