- FindStat

Identifier

Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000805: Integer compositions ⟶ ℤ

Values

[[1]] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1298 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q$

$F_{4} = 10\ q$

$F_{5} = 26\ q$

$F_{6} = 76\ q$

$F_{7} = 232\ q$

$F_{8} = 707\ q + 57\ q^{2}$

Description

The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

Map

peak composition

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

$T$ .