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Identifier

Mp00294: Standard tableaux —peak composition⟶ Integer compositions

Images

[] => []

[[1]] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2264 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

The composition corresponding to the peak set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a peak, if $i-1$ is an ascent and $i$ is a descent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of $T$.

References

[1] Zhou, R. D. P., Yan, S. H. F. Equidistribution of set-valued statistics on standard Young tableaux and transversals arXiv:2306.04382

Properties

graded

Sage code

def mapping(T):
    d = T.standard_descents()
    c = [i for i in range(2, T.size()) if i in d and i-1 not in d]
    return Composition(from_subset=(c, T.size()))

Weight

Created

Jun 12, 2023 at 07:45 by Martin Rubey

Updated

Jun 12, 2023 at 07:45 by Martin Rubey