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Identifier

St000898: Alternating sign matrices ⟶ ℤ

Values

[[1,0],[0,1]] => 1

[[0,1],[1,0]] => 2

[[1,0,0],[0,1,0],[0,0,1]] => 2

[[0,1,0],[1,0,0],[0,0,1]] => 3

[[1,0,0],[0,0,1],[0,1,0]] => 1

[[0,1,0],[1,-1,1],[0,1,0]] => 1

[[0,0,1],[1,0,0],[0,1,0]] => 2

[[0,1,0],[0,0,1],[1,0,0]] => 2

[[0,0,1],[0,1,0],[1,0,0]] => 3

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => 3

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 4

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => 2

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => 2

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => 3

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => 3

[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 4

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => 3

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => 4

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => 2

[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => 2

[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => 3

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => 3

[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => 4

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => 2

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => 2

[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => 2

[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => 3

[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => 3

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => 3

[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => 4

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => 1

[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => 1

[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => 2

[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => 2

[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => 3

[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => 1

[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => 1

[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => 1

[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => 2

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => 1

[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => 1

[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => 2

[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => 2

[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => 3

[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => 3

[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => 4

[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => 2

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => 2

[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => 3

[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => 3

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => 4

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 4

[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => 5

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 4

[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 5

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 3

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 3

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 5

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]] => 4

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Description

The number of maximal entries in the last diagonal of the monotone triangle.
Consider the alternating sign matrix
$$ \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right). $$
The corresponding monotone triangle is
$$ \begin{array}{ccccccccc} 5 & & 4 & & 3 & & 2 & & 1 \\ & 5 & & 4 & & 3 & & 1 & \\ & & 5 & & 3 & & 1 & & \\ & & & 5 & & 3 & & & \\ & & & & 4 & & & & \end{array} $$
The first entry $1$ in the last diagonal is maximal, because rows are strictly decreasing and its left neighbour is $2$. Also, the entry $3$ in the last diagonal is maximal, because diagonals from north-west to south-east are weakly decreasing, and its north-west neighbour is also $3$. All other entries in the last diagonal are non-maximal, thus the statistic on this matrix is $2$.
Conjecturally, this statistic is equidistributed with St000066The column of the unique '1' in the first row of the alternating sign matrix..

References

[1] Guoniu Han, private communication.

Code

def statistic(A):
    M = A.to_monotone_triangle()
    r = 0
    for i in range(len(M)):
        if ((i >= 1 and M[i][-1] == M[i-1][-2]) or
            (i < len(M)-1 and M[i][-1] == M[i][-2]-1)):
            r += 1
    return r

Created

Jul 18, 2017 at 00:03 by Martin Rubey

Updated

Jul 18, 2017 at 00:03 by Martin Rubey