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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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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2} + q^{3}$

$F_{4} = 1 + q + 2\ q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + q + 2\ q^{2} + 3\ q^{3} + 4\ q^{4} + 4\ q^{5} + 4\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + q + 2\ q^{2} + 4\ q^{3} + 5\ q^{4} + 7\ q^{5} + 9\ q^{6} + 9\ q^{7} + 9\ q^{8} + 9\ q^{9} + 7\ q^{10} + 5\ q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 9\ q^{5} + 13\ q^{6} + 16\ q^{7} + 19\ q^{8} + 22\ q^{9} + 23\ q^{10} + 23\ q^{11} + 22\ q^{12} + 19\ q^{13} + 16\ q^{14} + 13\ q^{15} + 9\ q^{16} + 6\ q^{17} + 4\ q^{18} + 2\ q^{19} + q^{20} + q^{21}$

$F_{8} = 1 + q + 2\ q^{2} + 4\ q^{3} + 7\ q^{4} + 10\ q^{5} + 16\ q^{6} + 22\ q^{7} + 30\ q^{8} + 37\ q^{9} + 46\ q^{10} + 52\ q^{11} + 60\ q^{12} + 62\ q^{13} + 64\ q^{14} + 62\ q^{15} + 60\ q^{16} + 52\ q^{17} + 46\ q^{18} + 37\ q^{19} + 30\ q^{20} + 22\ q^{21} + 16\ q^{22} + 10\ q^{23} + 7\ q^{24} + 4\ q^{25} + 2\ q^{26} + q^{27} + q^{28}$

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