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Identifier

St001697: Standard tableaux ⟶ ℤ

Values

[[1]] => 0

[[1,2]] => 0

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

[[1,2,3]] => 0

[[1,3],[2]] => 2

[[1,2],[3]] => 1

[[1],[2],[3]] => 3

[[1,2,3,4]] => 0

[[1,3,4],[2]] => 3

[[1,2,4],[3]] => 2

[[1,2,3],[4]] => 1

[[1,3],[2,4]] => 4

[[1,2],[3,4]] => 2

[[1,4],[2],[3]] => 4

[[1,3],[2],[4]] => 5

[[1,2],[3],[4]] => 3

[[1],[2],[3],[4]] => 6

[[1,2,3,4,5]] => 0

[[1,3,4,5],[2]] => 4

[[1,2,4,5],[3]] => 3

[[1,2,3,5],[4]] => 2

[[1,2,3,4],[5]] => 1

[[1,3,5],[2,4]] => 6

[[1,2,5],[3,4]] => 3

[[1,3,4],[2,5]] => 5

[[1,2,4],[3,5]] => 4

[[1,2,3],[4,5]] => 2

[[1,4,5],[2],[3]] => 5

[[1,3,5],[2],[4]] => 7

[[1,2,5],[3],[4]] => 4

[[1,3,4],[2],[5]] => 6

[[1,2,4],[3],[5]] => 5

[[1,2,3],[4],[5]] => 3

[[1,4],[2,5],[3]] => 6

[[1,3],[2,5],[4]] => 8

[[1,2],[3,5],[4]] => 5

[[1,3],[2,4],[5]] => 7

[[1,2],[3,4],[5]] => 4

[[1,5],[2],[3],[4]] => 7

[[1,4],[2],[3],[5]] => 8

[[1,3],[2],[4],[5]] => 9

[[1,2],[3],[4],[5]] => 6

[[1],[2],[3],[4],[5]] => 10

[[1,2,3,4,5,6]] => 0

[[1,3,4,5,6],[2]] => 5

[[1,2,4,5,6],[3]] => 4

[[1,2,3,5,6],[4]] => 3

[[1,2,3,4,6],[5]] => 2

[[1,2,3,4,5],[6]] => 1

[[1,3,5,6],[2,4]] => 8

[[1,2,5,6],[3,4]] => 4

[[1,3,4,6],[2,5]] => 7

[[1,2,4,6],[3,5]] => 6

[[1,2,3,6],[4,5]] => 3

[[1,3,4,5],[2,6]] => 6

[[1,2,4,5],[3,6]] => 5

[[1,2,3,5],[4,6]] => 4

[[1,2,3,4],[5,6]] => 2

[[1,4,5,6],[2],[3]] => 6

[[1,3,5,6],[2],[4]] => 9

[[1,2,5,6],[3],[4]] => 5

[[1,3,4,6],[2],[5]] => 8

[[1,2,4,6],[3],[5]] => 7

[[1,2,3,6],[4],[5]] => 4

[[1,3,4,5],[2],[6]] => 7

[[1,2,4,5],[3],[6]] => 6

[[1,2,3,5],[4],[6]] => 5

[[1,2,3,4],[5],[6]] => 3

[[1,3,5],[2,4,6]] => 9

[[1,2,5],[3,4,6]] => 5

[[1,3,4],[2,5,6]] => 7

[[1,2,4],[3,5,6]] => 6

[[1,2,3],[4,5,6]] => 3

[[1,4,6],[2,5],[3]] => 8

[[1,3,6],[2,5],[4]] => 11

[[1,2,6],[3,5],[4]] => 7

[[1,3,6],[2,4],[5]] => 9

[[1,2,6],[3,4],[5]] => 5

[[1,4,5],[2,6],[3]] => 7

[[1,3,5],[2,6],[4]] => 10

[[1,2,5],[3,6],[4]] => 6

[[1,3,4],[2,6],[5]] => 9

[[1,2,4],[3,6],[5]] => 8

[[1,2,3],[4,6],[5]] => 5

[[1,3,5],[2,4],[6]] => 10

[[1,2,5],[3,4],[6]] => 6

[[1,3,4],[2,5],[6]] => 8

[[1,2,4],[3,5],[6]] => 7

[[1,2,3],[4,5],[6]] => 4

[[1,5,6],[2],[3],[4]] => 8

[[1,4,6],[2],[3],[5]] => 10

[[1,3,6],[2],[4],[5]] => 11

[[1,2,6],[3],[4],[5]] => 7

[[1,4,5],[2],[3],[6]] => 9

[[1,3,5],[2],[4],[6]] => 12

[[1,2,5],[3],[4],[6]] => 8

[[1,3,4],[2],[5],[6]] => 10

[[1,2,4],[3],[5],[6]] => 9

[[1,2,3],[4],[5],[6]] => 6

[[1,4],[2,5],[3,6]] => 9

[[1,3],[2,5],[4,6]] => 12

>>> Load all 1115 entries. <<<

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Description

The shifted natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.

References

[1] Hopkins, S. Two majs for standard Young tableaux? MathOverflow:385374

Code

def natural_descents(T):
    n = T.size()
    D = []
    for i in range(1, n):
        for row in T:
            if row.count(i):
                break
            if row.count(i+1):
                D.append(i)
                break
    return D

def statistic(T):
    n = T.size()
    D = natural_descents(T)
    return sum(n-d for d in D) + sum(i*p for i, p in enumerate(T.shape()))

Created

Mar 04, 2021 at 08:53 by Martin Rubey

Updated

Mar 15, 2021 at 15:39 by Martin Rubey