Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000063: 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] => 8
[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] => 8
[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] => 8
[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] => 8
[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] => 10
[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] => 15
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 8
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Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
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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