Identifier
-
Mp00071:
Permutations
—descent composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000001: Permutations ⟶ ℤ
Values
[1] => [1] => [1,0] => [2,1] => 1
[1,2] => [2] => [1,1,0,0] => [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0] => [3,1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 9
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,5,3,1,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[3,1,4,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[3,1,5,2,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[3,1,5,4,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 9
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 6
[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 9
[3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[3,2,5,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 9
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[3,4,1,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[3,4,2,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[3,5,1,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
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Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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