- FindStat

Identifier

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000090: Integer compositions ⟶ ℤ

Values

[[1]] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

$F_{4} = 7 + 3\ q$

$F_{5} = 18 + 8\ q$

$F_{6} = 52 + 24\ q$

$F_{7} = 158 + 74\ q$

Description

The variation of a composition.

Map

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .

Map

valley composition

Description

The composition corresponding to the valley 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 valley if

$i-1$ is a descent and

$i$ is an ascent.
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 valley set of

$T$ .