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Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 13\ q + 8\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{6} = 38\ q + 24\ q^{2} + 9\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{7} = 116\ q + 74\ q^{2} + 28\ q^{3} + 9\ q^{4} + 3\ q^{5} + q^{6} + q^{7}$

Description

The last part of an integer 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

rotate front to back

Description

The front to back rotation of the entries of an integer composition.

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .