DecreasableCornerSumPositions on AlternatingSignMatrix
The interior positions of the corner sum matrix whose entry can be lowered by one to give another alternating sign matrix's corner sum matrix.
FindStat's St001965. The corner sum c(i, j) is the sum of the entries in the first i rows and j columns; a position 0 < i, j < n qualifies when c(i, j) = c(i + 1, j) = c(i, j + 1) = c(i - 1, j) + 1 = c(i, j - 1) + 1.
Called as CombinatorialStat(x, "DecreasableCornerSumPositions"). The head's page has its examples.
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Distribution
Over AlternatingSignMatrices at each size, tallied here in your browser; sizes past 6 are not offered.
FindStat St001965
The number of decreasable positions in the corner sum matrix of an alternating sign matrix.
| Object | Value |
|---|---|
[[1]] | 0 |
[[0,1],[1,0]] | 0 |
[[1,0],[0,1]] | 1 |
[[0,0,1],[0,1,0],[1,0,0]] | 0 |
[[0,0,1],[1,0,0],[0,1,0]] | 1 |
[[0,1,0],[0,0,1],[1,0,0]] | 1 |
[[0,1,0],[1,0,0],[0,0,1]] | 1 |
[[1,0,0],[0,0,1],[0,1,0]] | 1 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001965 has the rest.