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Identifier

St000888: Alternating sign matrices ⟶ ℤ

Values

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

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

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

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

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

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

[[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]] => 1

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

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

[[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]] => 2

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

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

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

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

[[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]] => 3

[[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]] => 1

[[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]] => 3

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

[[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]] => 1

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

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

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

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

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

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

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

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

[[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]] => 2

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

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

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

[[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]] => 2

[[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]] => 1

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

[[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]] => 2

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

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

[[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]] => 5

[[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]] => 3

[[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]] => 2

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 3

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 3

[[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]] => 1

[[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]] => 2

[[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]] => 3

[[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]] => 1

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 1

[[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]] => 2

[[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]] => 3

[[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]] => 1

[[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]] => 2

[[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]] => 3

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 2

[[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]] => 1

[[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]] => 3

[[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]] => 2

[[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]] => 2

[[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]] => 3

[[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]] => 3

[[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]] => 1

[[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]] => 2

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Description

The maximal sum of entries on a diagonal of an alternating sign matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$$
are $(0,1,1,0,1,1,0)$, so the statistic is $1$.
This is a natural extension of St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. to alternating sign matrices.

Code

def statistic(pi):
    P = pi.to_matrix()
    n = P.nrows()
    return max(sum(P[n-k+j][j] for j in range(max(0, k-n), min(k, n)))
               for k in range(1, 2*n))

Created

Jul 14, 2017 at 12:02 by Martin Rubey

Updated

Oct 23, 2017 at 12:02 by Martin Rubey