Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001400: Integer partitions ⟶ ℤ
Values
[1] => [[1],[]] => [] => 1
[1,1] => [[1,1],[]] => [] => 1
[2] => [[2],[]] => [] => 1
[1,1,1] => [[1,1,1],[]] => [] => 1
[1,2] => [[2,1],[]] => [] => 1
[2,1] => [[2,2],[1]] => [1] => 2
[3] => [[3],[]] => [] => 1
[1,1,1,1] => [[1,1,1,1],[]] => [] => 1
[1,1,2] => [[2,1,1],[]] => [] => 1
[1,2,1] => [[2,2,1],[1]] => [1] => 2
[1,3] => [[3,1],[]] => [] => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => 3
[2,2] => [[3,2],[1]] => [1] => 2
[3,1] => [[3,3],[2]] => [2] => 3
[4] => [[4],[]] => [] => 1
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [] => 1
[1,1,1,2] => [[2,1,1,1],[]] => [] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => 2
[1,1,3] => [[3,1,1],[]] => [] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 3
[1,2,2] => [[3,2,1],[1]] => [1] => 2
[1,3,1] => [[3,3,1],[2]] => [2] => 3
[1,4] => [[4,1],[]] => [] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 4
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => 3
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 6
[2,3] => [[4,2],[1]] => [1] => 2
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => 6
[3,2] => [[4,3],[2]] => [2] => 3
[4,1] => [[4,4],[3]] => [3] => 4
[5] => [[5],[]] => [] => 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]] => [] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 2
[1,1,1,3] => [[3,1,1,1],[]] => [] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 3
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => 2
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => 3
[1,1,4] => [[4,1,1],[]] => [] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 4
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 3
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 6
[1,2,3] => [[4,2,1],[1]] => [1] => 2
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 6
[1,3,2] => [[4,3,1],[2]] => [2] => 3
[1,4,1] => [[4,4,1],[3]] => [3] => 4
[1,5] => [[5,1],[]] => [] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 4
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 9
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => 3
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 12
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => 6
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 9
[2,4] => [[5,2],[1]] => [1] => 2
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 10
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => 6
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 12
[3,3] => [[5,3],[2]] => [2] => 3
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 10
[4,2] => [[5,4],[3]] => [3] => 4
[5,1] => [[5,5],[4]] => [4] => 5
[6] => [[6],[]] => [] => 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 2
[1,1,1,1,3] => [[3,1,1,1,1],[]] => [] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 3
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 3
[1,1,1,4] => [[4,1,1,1],[]] => [] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 4
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 6
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 6
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => 3
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => 4
[1,1,5] => [[5,1,1],[]] => [] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 5
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 4
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 9
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 3
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 12
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 6
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 9
[1,2,4] => [[5,2,1],[1]] => [1] => 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 10
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 6
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 12
[1,3,3] => [[5,3,1],[2]] => [2] => 3
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 10
[1,4,2] => [[5,4,1],[3]] => [3] => 4
[1,5,1] => [[5,5,1],[4]] => [4] => 5
[1,6] => [[6,1],[]] => [] => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 6
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 5
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 12
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 4
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 18
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 9
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Description
The total number of Littlewood-Richardson tableaux of given shape.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.
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.
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.
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