Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000001: Permutations ⟶ ℤ
Values
[1] => [1,0,1,0] => [1,1,0,0] => [1,2] => 1
[2] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 1
[1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,3,2] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [3,1,2] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 1
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 1
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,6,2,4,5] => 3
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => 1
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [7,1,2,3,4,5,6] => 1
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => 1
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 1
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 5
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 3
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 1
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,4,5] => 3
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 2
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => 2
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [1,3,7,2,4,5,6] => 4
[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,3,2,4,5,6,7,8] => 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9] => 1
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,7,2,3,4,5,6] => 1
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5,7] => 1
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => 1
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 14
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 5
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 5
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 3
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 3
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,3,5,6,2,4] => 5
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 1
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 1
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9,10] => 1
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [6,7,1,2,3,4,5] => 42
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [5,1,2,3,4,6,7] => 1
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => 1
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => 14
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => 1
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => 14
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => 1
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,6,4] => 4
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 5
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => 5
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 1
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 5
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 2
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,2,4,6,3,5] => 2
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,3,4,6,2,5] => 3
[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,3,2,8,4,5,6,7] => 5
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [3,1,4,2,5,6] => 2
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [4,1,2,3,5,6,7] => 1
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 1
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => 9
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => 5
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 14
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => 5
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => 9
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => 3
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,6,5] => 4
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 5
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 3
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 2
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 3
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,1,2,5,3,6] => 3
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 1
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 1
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [3,1,4,6,2,5] => 11
[7,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [1,2,3,8,4,5,6,7] => 1
[7,2,2] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [1,7,2,3,4,5,6,8] => 1
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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
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.
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