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Identifier
Values
[[1]] => 0
[[2]] => 1
[[1,1]] => 0
[[1,2]] => 1
[[2,2]] => 2
[[1],[2]] => 0
[[1,3]] => 2
[[2,3]] => 3
[[3,3]] => 4
[[1],[3]] => 1
[[2],[3]] => 2
[[1,4]] => 3
[[2,4]] => 4
[[3,4]] => 5
[[4,4]] => 6
[[1],[4]] => 2
[[2],[4]] => 3
[[3],[4]] => 4
[[1,5]] => 4
[[2,5]] => 5
[[3,5]] => 6
[[4,5]] => 7
[[5,5]] => 8
[[1],[5]] => 3
[[2],[5]] => 4
[[3],[5]] => 5
[[4],[5]] => 6
[[1,6]] => 5
[[2,6]] => 6
[[3,6]] => 7
[[4,6]] => 8
[[5,6]] => 9
[[6,6]] => 10
[[1],[6]] => 4
[[2],[6]] => 5
[[3],[6]] => 6
[[4],[6]] => 7
[[5],[6]] => 8
[[1,1,1]] => 0
[[1,1,2]] => 1
[[1,2,2]] => 2
[[2,2,2]] => 3
[[1,1],[2]] => 0
[[1,2],[2]] => 1
[[1,1,3]] => 2
[[1,2,3]] => 3
[[1,3,3]] => 4
[[2,2,3]] => 4
[[2,3,3]] => 5
[[3,3,3]] => 6
[[1,1],[3]] => 1
[[1,2],[3]] => 2
[[1,3],[2]] => 2
[[1,3],[3]] => 3
[[2,2],[3]] => 3
[[2,3],[3]] => 4
[[1],[2],[3]] => 0
[[1,1,4]] => 3
[[1,2,4]] => 4
[[1,3,4]] => 5
[[1,4,4]] => 6
[[2,2,4]] => 5
[[2,3,4]] => 6
[[2,4,4]] => 7
[[3,3,4]] => 7
[[3,4,4]] => 8
[[4,4,4]] => 9
[[1,1],[4]] => 2
[[1,2],[4]] => 3
[[1,4],[2]] => 3
[[1,3],[4]] => 4
[[1,4],[3]] => 4
[[1,4],[4]] => 5
[[2,2],[4]] => 4
[[2,3],[4]] => 5
[[2,4],[3]] => 5
[[2,4],[4]] => 6
[[3,3],[4]] => 6
[[3,4],[4]] => 7
[[1],[2],[4]] => 1
[[1],[3],[4]] => 2
[[2],[3],[4]] => 3
[[1,1,5]] => 4
[[1,2,5]] => 5
[[1,3,5]] => 6
[[1,4,5]] => 7
[[1,5,5]] => 8
[[2,2,5]] => 6
[[2,3,5]] => 7
[[2,4,5]] => 8
[[2,5,5]] => 9
[[3,3,5]] => 8
[[3,4,5]] => 9
[[3,5,5]] => 10
[[4,4,5]] => 10
[[4,5,5]] => 11
[[5,5,5]] => 12
[[1,1],[5]] => 3
[[1,2],[5]] => 4
[[1,5],[2]] => 4
[[1,3],[5]] => 5
>>> Load all 322 entries. <<<
[[1,5],[3]] => 5
[[1,4],[5]] => 6
[[1,5],[4]] => 6
[[1,5],[5]] => 7
[[2,2],[5]] => 5
[[2,3],[5]] => 6
[[2,5],[3]] => 6
[[2,4],[5]] => 7
[[2,5],[4]] => 7
[[2,5],[5]] => 8
[[3,3],[5]] => 7
[[3,4],[5]] => 8
[[3,5],[4]] => 8
[[3,5],[5]] => 9
[[4,4],[5]] => 9
[[4,5],[5]] => 10
[[1],[2],[5]] => 2
[[1],[3],[5]] => 3
[[1],[4],[5]] => 4
[[2],[3],[5]] => 4
[[2],[4],[5]] => 5
[[3],[4],[5]] => 6
[[1,1,1,1]] => 0
[[1,1,1,2]] => 1
[[1,1,2,2]] => 2
[[1,2,2,2]] => 3
[[2,2,2,2]] => 4
[[1,1,1],[2]] => 0
[[1,1,2],[2]] => 1
[[1,2,2],[2]] => 2
[[1,1],[2,2]] => 0
[[1,1,1,3]] => 2
[[1,1,2,3]] => 3
[[1,1,3,3]] => 4
[[1,2,2,3]] => 4
[[1,2,3,3]] => 5
[[1,3,3,3]] => 6
[[2,2,2,3]] => 5
[[2,2,3,3]] => 6
[[2,3,3,3]] => 7
[[3,3,3,3]] => 8
[[1,1,1],[3]] => 1
[[1,1,2],[3]] => 2
[[1,1,3],[2]] => 2
[[1,1,3],[3]] => 3
[[1,2,2],[3]] => 3
[[1,2,3],[2]] => 3
[[1,2,3],[3]] => 4
[[1,3,3],[2]] => 4
[[1,3,3],[3]] => 5
[[2,2,2],[3]] => 4
[[2,2,3],[3]] => 5
[[2,3,3],[3]] => 6
[[1,1],[2,3]] => 1
[[1,1],[3,3]] => 2
[[1,2],[2,3]] => 2
[[1,2],[3,3]] => 3
[[2,2],[3,3]] => 4
[[1,1],[2],[3]] => 0
[[1,2],[2],[3]] => 1
[[1,3],[2],[3]] => 2
[[1,1,1,4]] => 3
[[1,1,2,4]] => 4
[[1,1,3,4]] => 5
[[1,1,4,4]] => 6
[[1,2,2,4]] => 5
[[1,2,3,4]] => 6
[[1,2,4,4]] => 7
[[1,3,3,4]] => 7
[[1,3,4,4]] => 8
[[1,4,4,4]] => 9
[[2,2,2,4]] => 6
[[2,2,3,4]] => 7
[[2,2,4,4]] => 8
[[2,3,3,4]] => 8
[[2,3,4,4]] => 9
[[2,4,4,4]] => 10
[[3,3,3,4]] => 9
[[3,3,4,4]] => 10
[[3,4,4,4]] => 11
[[4,4,4,4]] => 12
[[1,1,1],[4]] => 2
[[1,1,2],[4]] => 3
[[1,1,4],[2]] => 3
[[1,1,3],[4]] => 4
[[1,1,4],[3]] => 4
[[1,1,4],[4]] => 5
[[1,2,2],[4]] => 4
[[1,2,4],[2]] => 4
[[1,2,3],[4]] => 5
[[1,2,4],[3]] => 5
[[1,3,4],[2]] => 5
[[1,2,4],[4]] => 6
[[1,4,4],[2]] => 6
[[1,3,3],[4]] => 6
[[1,3,4],[3]] => 6
[[1,3,4],[4]] => 7
[[1,4,4],[3]] => 7
[[1,4,4],[4]] => 8
[[2,2,2],[4]] => 5
[[2,2,3],[4]] => 6
[[2,2,4],[3]] => 6
[[2,2,4],[4]] => 7
[[2,3,3],[4]] => 7
[[2,3,4],[3]] => 7
[[2,3,4],[4]] => 8
[[2,4,4],[3]] => 8
[[2,4,4],[4]] => 9
[[3,3,3],[4]] => 8
[[3,3,4],[4]] => 9
[[3,4,4],[4]] => 10
[[1,1],[2,4]] => 2
[[1,1],[3,4]] => 3
[[1,1],[4,4]] => 4
[[1,2],[2,4]] => 3
[[1,2],[3,4]] => 4
[[1,3],[2,4]] => 4
[[1,2],[4,4]] => 5
[[1,3],[3,4]] => 5
[[1,3],[4,4]] => 6
[[2,2],[3,4]] => 5
[[2,2],[4,4]] => 6
[[2,3],[3,4]] => 6
[[2,3],[4,4]] => 7
[[3,3],[4,4]] => 8
[[1,1],[2],[4]] => 1
[[1,1],[3],[4]] => 2
[[1,2],[2],[4]] => 2
[[1,2],[3],[4]] => 3
[[1,3],[2],[4]] => 3
[[1,4],[2],[3]] => 3
[[1,4],[2],[4]] => 4
[[1,3],[3],[4]] => 4
[[1,4],[3],[4]] => 5
[[2,2],[3],[4]] => 4
[[2,3],[3],[4]] => 5
[[2,4],[3],[4]] => 6
[[1],[2],[3],[4]] => 0
[[1,1,1,1,2]] => 1
[[1,1,1,2,2]] => 2
[[1,1,2,2,2]] => 3
[[1,2,2,2,2]] => 4
[[2,2,2,2,2]] => 5
[[1,1,1,1],[2]] => 0
[[1,1,1,2],[2]] => 1
[[1,1,2,2],[2]] => 2
[[1,2,2,2],[2]] => 3
[[1,1,1],[2,2]] => 0
[[1,1,2],[2,2]] => 1
[[1,1,1,1,3]] => 2
[[1,1,1,2,3]] => 3
[[1,1,1,3,3]] => 4
[[1,1,2,2,3]] => 4
[[1,1,2,3,3]] => 5
[[1,1,3,3,3]] => 6
[[1,2,2,2,3]] => 5
[[1,2,2,3,3]] => 6
[[1,2,3,3,3]] => 7
[[1,3,3,3,3]] => 8
[[2,2,2,2,3]] => 6
[[2,2,2,3,3]] => 7
[[2,2,3,3,3]] => 8
[[2,3,3,3,3]] => 9
[[3,3,3,3,3]] => 10
[[1,1,1,1],[3]] => 1
[[1,1,1,2],[3]] => 2
[[1,1,1,3],[2]] => 2
[[1,1,1,3],[3]] => 3
[[1,1,2,2],[3]] => 3
[[1,1,2,3],[2]] => 3
[[1,1,2,3],[3]] => 4
[[1,1,3,3],[2]] => 4
[[1,1,3,3],[3]] => 5
[[1,2,2,2],[3]] => 4
[[1,2,2,3],[2]] => 4
[[1,2,2,3],[3]] => 5
[[1,2,3,3],[2]] => 5
[[1,2,3,3],[3]] => 6
[[1,3,3,3],[2]] => 6
[[1,3,3,3],[3]] => 7
[[2,2,2,2],[3]] => 5
[[2,2,2,3],[3]] => 6
[[2,2,3,3],[3]] => 7
[[2,3,3,3],[3]] => 8
[[1,1,1],[2,3]] => 1
[[1,1,1],[3,3]] => 2
[[1,1,2],[2,3]] => 2
[[1,1,3],[2,2]] => 2
[[1,1,2],[3,3]] => 3
[[1,1,3],[2,3]] => 3
[[1,1,3],[3,3]] => 4
[[1,2,2],[2,3]] => 3
[[1,2,2],[3,3]] => 4
[[1,2,3],[2,3]] => 4
[[1,2,3],[3,3]] => 5
[[2,2,2],[3,3]] => 5
[[2,2,3],[3,3]] => 6
[[1,1,1],[2],[3]] => 0
[[1,1,2],[2],[3]] => 1
[[1,1,3],[2],[3]] => 2
[[1,2,2],[2],[3]] => 2
[[1,2,3],[2],[3]] => 3
[[1,3,3],[2],[3]] => 4
[[1,1],[2,2],[3]] => 0
[[1,1],[2,3],[3]] => 1
[[1,2],[2,3],[3]] => 2
[[1,1,1,1,1,2]] => 1
[[1,1,1,1,2,2]] => 2
[[1,1,1,2,2,2]] => 3
[[1,1,2,2,2,2]] => 4
[[1,2,2,2,2,2]] => 5
[[2,2,2,2,2,2]] => 6
[[1,1,1,1,1],[2]] => 0
[[1,1,1,1,2],[2]] => 1
[[1,1,1,2,2],[2]] => 2
[[1,1,2,2,2],[2]] => 3
[[1,2,2,2,2],[2]] => 4
[[1,1,1,1],[2,2]] => 0
[[1,1,1,2],[2,2]] => 1
[[1,1,2,2],[2,2]] => 2
[[1,1,1],[2,2,2]] => 0
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Description
The sum of the entries reduced by the index of their row in a semistandard tableau.
This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.
Code
def statistic(T):
    return sum(e-i for i, row in enumerate(T, 1) for e in row)

def statistic(T):
    la = T.shape()
    n = max(T.entries())-1
    C = crystals.Tableaux(CartanType(["A", n]), shape=la)
    w = C(rows=T)
    return len(w.to_highest_weight()[1])
Created
Jun 15, 2013 at 15:48 by Travis Scrimshaw
Updated
Feb 21, 2021 at 14:57 by Martin Rubey