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Identifier
Values
=>
Cc0019;cc-rep
[[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 [[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