- FindStat

Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000512: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,5,6,7],[4]] => [3,4] => [[6,3],[2]] => [2] => 0

[[1,2,3,4,6,7],[5]] => [4,3] => [[6,4],[3]] => [3] => 1

[[1,2,3,4,5,7],[6]] => [5,2] => [[6,5],[4]] => [4] => 0

[[1,2,3,4,5,6],[7]] => [6,1] => [[6,6],[5]] => [5] => 0

[[1,3,4,6,7],[2,5]] => [1,3,3] => [[5,3,1],[2]] => [2] => 0

[[1,2,4,6,7],[3,5]] => [2,2,3] => [[5,3,2],[2,1]] => [2,1] => 1

[[1,2,3,6,7],[4,5]] => [3,4] => [[6,3],[2]] => [2] => 0

[[1,3,4,5,7],[2,6]] => [1,4,2] => [[5,4,1],[3]] => [3] => 1

[[1,2,4,5,7],[3,6]] => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 1

[[1,2,3,5,7],[4,6]] => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 1

[[1,2,3,4,7],[5,6]] => [4,3] => [[6,4],[3]] => [3] => 1

[[1,3,4,5,6],[2,7]] => [1,5,1] => [[5,5,1],[4]] => [4] => 0

[[1,2,4,5,6],[3,7]] => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 0

[[1,2,3,5,6],[4,7]] => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 0

[[1,2,3,4,6],[5,7]] => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 1

[[1,2,3,4,5],[6,7]] => [5,2] => [[6,5],[4]] => [4] => 0

[[1,2,5,6,7],[3],[4]] => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 0

[[1,3,4,6,7],[2],[5]] => [1,3,3] => [[5,3,1],[2]] => [2] => 0

[[1,2,4,6,7],[3],[5]] => [2,2,3] => [[5,3,2],[2,1]] => [2,1] => 1

[[1,2,3,6,7],[4],[5]] => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 0

[[1,3,4,5,7],[2],[6]] => [1,4,2] => [[5,4,1],[3]] => [3] => 1

[[1,2,4,5,7],[3],[6]] => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 1

[[1,2,3,5,7],[4],[6]] => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 1

[[1,2,3,4,7],[5],[6]] => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 2

[[1,3,4,5,6],[2],[7]] => [1,5,1] => [[5,5,1],[4]] => [4] => 0

[[1,2,4,5,6],[3],[7]] => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 0

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Description

The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.

Map

inner shape

Description

The inner shape of a skew partition.

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

$i$ .