Identifier
-
Mp00071:
Permutations
—descent composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001487: Skew partitions ⟶ ℤ
Values
[1] => [1] => [[1],[]] => 1
[1,2] => [2] => [[2],[]] => 1
[2,1] => [1,1] => [[1,1],[]] => 1
[1,2,3] => [3] => [[3],[]] => 1
[1,3,2] => [2,1] => [[2,2],[1]] => 2
[2,1,3] => [1,2] => [[2,1],[]] => 1
[2,3,1] => [2,1] => [[2,2],[1]] => 2
[3,1,2] => [1,2] => [[2,1],[]] => 1
[3,2,1] => [1,1,1] => [[1,1,1],[]] => 1
[1,2,3,4] => [4] => [[4],[]] => 1
[1,2,4,3] => [3,1] => [[3,3],[2]] => 2
[1,3,2,4] => [2,2] => [[3,2],[1]] => 2
[1,3,4,2] => [3,1] => [[3,3],[2]] => 2
[1,4,2,3] => [2,2] => [[3,2],[1]] => 2
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]] => 2
[2,1,3,4] => [1,3] => [[3,1],[]] => 1
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]] => 2
[2,3,1,4] => [2,2] => [[3,2],[1]] => 2
[2,3,4,1] => [3,1] => [[3,3],[2]] => 2
[2,4,1,3] => [2,2] => [[3,2],[1]] => 2
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]] => 2
[3,1,2,4] => [1,3] => [[3,1],[]] => 1
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]] => 2
[3,2,1,4] => [1,1,2] => [[2,1,1],[]] => 1
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]] => 2
[3,4,1,2] => [2,2] => [[3,2],[1]] => 2
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 2
[4,1,2,3] => [1,3] => [[3,1],[]] => 1
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]] => 2
[4,2,1,3] => [1,1,2] => [[2,1,1],[]] => 1
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]] => 2
[4,3,1,2] => [1,1,2] => [[2,1,1],[]] => 1
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]] => 1
[1,2,3,4,5] => [5] => [[5],[]] => 1
[1,2,3,5,4] => [4,1] => [[4,4],[3]] => 2
[1,2,4,3,5] => [3,2] => [[4,3],[2]] => 2
[1,2,4,5,3] => [4,1] => [[4,4],[3]] => 2
[1,2,5,3,4] => [3,2] => [[4,3],[2]] => 2
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]] => 2
[1,3,2,4,5] => [2,3] => [[4,2],[1]] => 2
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]] => 3
[1,3,4,2,5] => [3,2] => [[4,3],[2]] => 2
[1,3,4,5,2] => [4,1] => [[4,4],[3]] => 2
[1,3,5,2,4] => [3,2] => [[4,3],[2]] => 2
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]] => 2
[1,4,2,3,5] => [2,3] => [[4,2],[1]] => 2
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]] => 3
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]] => 2
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
[1,4,5,2,3] => [3,2] => [[4,3],[2]] => 2
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]] => 2
[1,5,2,3,4] => [2,3] => [[4,2],[1]] => 2
[1,5,2,4,3] => [2,2,1] => [[3,3,2],[2,1]] => 3
[1,5,3,2,4] => [2,1,2] => [[3,2,2],[1,1]] => 2
[1,5,3,4,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
[1,5,4,2,3] => [2,1,2] => [[3,2,2],[1,1]] => 2
[1,5,4,3,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 2
[2,1,3,4,5] => [1,4] => [[4,1],[]] => 1
[2,1,3,5,4] => [1,3,1] => [[3,3,1],[2]] => 2
[2,1,4,3,5] => [1,2,2] => [[3,2,1],[1]] => 2
[2,1,4,5,3] => [1,3,1] => [[3,3,1],[2]] => 2
[2,1,5,3,4] => [1,2,2] => [[3,2,1],[1]] => 2
[2,1,5,4,3] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
[2,3,1,4,5] => [2,3] => [[4,2],[1]] => 2
[2,3,1,5,4] => [2,2,1] => [[3,3,2],[2,1]] => 3
[2,3,4,1,5] => [3,2] => [[4,3],[2]] => 2
[2,3,4,5,1] => [4,1] => [[4,4],[3]] => 2
[2,3,5,1,4] => [3,2] => [[4,3],[2]] => 2
[2,3,5,4,1] => [3,1,1] => [[3,3,3],[2,2]] => 2
[2,4,1,3,5] => [2,3] => [[4,2],[1]] => 2
[2,4,1,5,3] => [2,2,1] => [[3,3,2],[2,1]] => 3
[2,4,3,1,5] => [2,1,2] => [[3,2,2],[1,1]] => 2
[2,4,3,5,1] => [2,2,1] => [[3,3,2],[2,1]] => 3
[2,4,5,1,3] => [3,2] => [[4,3],[2]] => 2
[2,4,5,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 2
[2,5,1,3,4] => [2,3] => [[4,2],[1]] => 2
[2,5,1,4,3] => [2,2,1] => [[3,3,2],[2,1]] => 3
[2,5,3,1,4] => [2,1,2] => [[3,2,2],[1,1]] => 2
[2,5,3,4,1] => [2,2,1] => [[3,3,2],[2,1]] => 3
[2,5,4,1,3] => [2,1,2] => [[3,2,2],[1,1]] => 2
[2,5,4,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 2
[3,1,2,4,5] => [1,4] => [[4,1],[]] => 1
[3,1,2,5,4] => [1,3,1] => [[3,3,1],[2]] => 2
[3,1,4,2,5] => [1,2,2] => [[3,2,1],[1]] => 2
[3,1,4,5,2] => [1,3,1] => [[3,3,1],[2]] => 2
[3,1,5,2,4] => [1,2,2] => [[3,2,1],[1]] => 2
[3,1,5,4,2] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
[3,2,1,4,5] => [1,1,3] => [[3,1,1],[]] => 1
[3,2,1,5,4] => [1,1,2,1] => [[2,2,1,1],[1]] => 2
[3,2,4,1,5] => [1,2,2] => [[3,2,1],[1]] => 2
[3,2,4,5,1] => [1,3,1] => [[3,3,1],[2]] => 2
[3,2,5,1,4] => [1,2,2] => [[3,2,1],[1]] => 2
[3,2,5,4,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
[3,4,1,2,5] => [2,3] => [[4,2],[1]] => 2
[3,4,1,5,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
[3,4,2,1,5] => [2,1,2] => [[3,2,2],[1,1]] => 2
[3,4,2,5,1] => [2,2,1] => [[3,3,2],[2,1]] => 3
[3,4,5,1,2] => [3,2] => [[4,3],[2]] => 2
[3,4,5,2,1] => [3,1,1] => [[3,3,3],[2,2]] => 2
[3,5,1,2,4] => [2,3] => [[4,2],[1]] => 2
[3,5,1,4,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
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Description
The number of inner corners of a skew partition.
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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.
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