Identifier
-
Mp00071:
Permutations
—descent composition⟶
Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤ
Values
[1] => [1] => [1] => [[1],[]] => 1
[1,2] => [2] => [2] => [[2],[]] => 2
[2,1] => [1,1] => [1,1] => [[1,1],[]] => 2
[1,2,3] => [3] => [3] => [[3],[]] => 2
[1,3,2] => [2,1] => [2,1] => [[2,2],[1]] => 3
[2,1,3] => [1,2] => [1,2] => [[2,1],[]] => 3
[2,3,1] => [2,1] => [2,1] => [[2,2],[1]] => 3
[3,1,2] => [1,2] => [1,2] => [[2,1],[]] => 3
[3,2,1] => [1,1,1] => [1,1,1] => [[1,1,1],[]] => 2
[1,2,3,4] => [4] => [4] => [[4],[]] => 2
[1,2,4,3] => [3,1] => [3,1] => [[3,3],[2]] => 3
[1,3,2,4] => [2,2] => [2,2] => [[3,2],[1]] => 4
[1,3,4,2] => [3,1] => [3,1] => [[3,3],[2]] => 3
[1,4,2,3] => [2,2] => [2,2] => [[3,2],[1]] => 4
[1,4,3,2] => [2,1,1] => [1,2,1] => [[2,2,1],[1]] => 4
[2,1,3,4] => [1,3] => [1,3] => [[3,1],[]] => 3
[2,1,4,3] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 3
[2,3,1,4] => [2,2] => [2,2] => [[3,2],[1]] => 4
[2,3,4,1] => [3,1] => [3,1] => [[3,3],[2]] => 3
[2,4,1,3] => [2,2] => [2,2] => [[3,2],[1]] => 4
[2,4,3,1] => [2,1,1] => [1,2,1] => [[2,2,1],[1]] => 4
[3,1,2,4] => [1,3] => [1,3] => [[3,1],[]] => 3
[3,1,4,2] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 3
[3,2,1,4] => [1,1,2] => [1,1,2] => [[2,1,1],[]] => 3
[3,2,4,1] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 3
[3,4,1,2] => [2,2] => [2,2] => [[3,2],[1]] => 4
[3,4,2,1] => [2,1,1] => [1,2,1] => [[2,2,1],[1]] => 4
[4,1,2,3] => [1,3] => [1,3] => [[3,1],[]] => 3
[4,1,3,2] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 3
[4,2,1,3] => [1,1,2] => [1,1,2] => [[2,1,1],[]] => 3
[4,2,3,1] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => 3
[4,3,1,2] => [1,1,2] => [1,1,2] => [[2,1,1],[]] => 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]] => 2
[1,2,3,4,5] => [5] => [5] => [[5],[]] => 2
[1,2,3,5,4] => [4,1] => [4,1] => [[4,4],[3]] => 3
[1,2,4,3,5] => [3,2] => [3,2] => [[4,3],[2]] => 4
[1,2,4,5,3] => [4,1] => [4,1] => [[4,4],[3]] => 3
[1,2,5,3,4] => [3,2] => [3,2] => [[4,3],[2]] => 4
[1,2,5,4,3] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[1,3,2,4,5] => [2,3] => [2,3] => [[4,2],[1]] => 4
[1,3,2,5,4] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[1,3,4,2,5] => [3,2] => [3,2] => [[4,3],[2]] => 4
[1,3,4,5,2] => [4,1] => [4,1] => [[4,4],[3]] => 3
[1,3,5,2,4] => [3,2] => [3,2] => [[4,3],[2]] => 4
[1,3,5,4,2] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[1,4,2,3,5] => [2,3] => [2,3] => [[4,2],[1]] => 4
[1,4,2,5,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[1,4,3,2,5] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[1,4,3,5,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[1,4,5,2,3] => [3,2] => [3,2] => [[4,3],[2]] => 4
[1,4,5,3,2] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[1,5,2,3,4] => [2,3] => [2,3] => [[4,2],[1]] => 4
[1,5,2,4,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[1,5,3,2,4] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[1,5,3,4,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[1,5,4,2,3] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]] => 4
[2,1,3,4,5] => [1,4] => [1,4] => [[4,1],[]] => 3
[2,1,3,5,4] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 3
[2,1,4,3,5] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[2,1,4,5,3] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 3
[2,1,5,3,4] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
[2,3,1,4,5] => [2,3] => [2,3] => [[4,2],[1]] => 4
[2,3,1,5,4] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[2,3,4,1,5] => [3,2] => [3,2] => [[4,3],[2]] => 4
[2,3,4,5,1] => [4,1] => [4,1] => [[4,4],[3]] => 3
[2,3,5,1,4] => [3,2] => [3,2] => [[4,3],[2]] => 4
[2,3,5,4,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[2,4,1,3,5] => [2,3] => [2,3] => [[4,2],[1]] => 4
[2,4,1,5,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[2,4,3,1,5] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[2,4,3,5,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[2,4,5,1,3] => [3,2] => [3,2] => [[4,3],[2]] => 4
[2,4,5,3,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[2,5,1,3,4] => [2,3] => [2,3] => [[4,2],[1]] => 4
[2,5,1,4,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[2,5,3,1,4] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[2,5,3,4,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[2,5,4,1,3] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]] => 4
[3,1,2,4,5] => [1,4] => [1,4] => [[4,1],[]] => 3
[3,1,2,5,4] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 3
[3,1,4,2,5] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[3,1,4,5,2] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 3
[3,1,5,2,4] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
[3,2,1,4,5] => [1,1,3] => [1,1,3] => [[3,1,1],[]] => 3
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 3
[3,2,4,1,5] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[3,2,4,5,1] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => 3
[3,2,5,1,4] => [1,2,2] => [1,2,2] => [[3,2,1],[1]] => 5
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
[3,4,1,2,5] => [2,3] => [2,3] => [[4,2],[1]] => 4
[3,4,1,5,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[3,4,2,1,5] => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 4
[3,4,2,5,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
[3,4,5,1,2] => [3,2] => [3,2] => [[4,3],[2]] => 4
[3,4,5,2,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]] => 4
[3,5,1,2,4] => [2,3] => [2,3] => [[4,2],[1]] => 4
[3,5,1,4,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
This is also known as the number of removable cells of the 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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.
See Mp00314Foata bijection.
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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