- FindStat

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Identifier

St001816: Standard tableaux ⟶ ℤ

Values

[] => 0

[[1]] => 1

[[1,2]] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 352 entries. <<<

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Description

Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..

References

[1] Garsia, A. M., Wallach, N. $r$-Qsym is free over Sym MathSciNet:2319171
[2] Lafrenière, N. Eigenvalues of symmetrized shuffling operators MathSciNet:4098299
[3] Reiner, V., Saliola, F., Welker, V. Spectra of symmetrized shuffling operators MathSciNet:3184410

Code

def statistic(T):
    if T.height() <= 1:
        return T.size()
    r = 0
    for i in range(1, T.height()-1):
        if T[i+1][0] != T[i][0]+1:
            break
        r += 1
    j = T[1][0]-2 if r % 2 == 0 else T[1][0]-1
    return j

Created

Jul 13, 2022 at 17:16 by Nadia Lafreniere

Updated

Jul 13, 2022 at 17:16 by Nadia Lafreniere