- FindStat

Your data matches 52 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000488
(load all 14 compositions to match this statistic)

Mapping

St000488: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 3
[4,3,1,2] => 0
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 4
[1,4,3,5,2] => 2
[1,4,5,2,3] => 3
[1,4,5,3,2] => 1

Description

The number of cycles of a permutation of length at most 2.

Matching statistic: St000153

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 2
[2,1] => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 3
[1,3,2] => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 2
[3,1,2] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 0
[3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2

Description

The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).

Matching statistic: St000239

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000239: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 2
[2,1] => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 3
[1,3,2] => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 2
[3,1,2] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 0
[3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 2

Description

The number of small weak excedances. A small weak excedance is an index

$i$ such that

$\pi_i \in \{i,i+1\}$ . This is the sum of [[St000022]] and [[St000237]].

Matching statistic: St000620
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 66%●distinct values known / distinct values provided: 43%

Values

[1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {1,2}
[2,1] => [2,1] => [2]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,3,1,4] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[2,3,5,1,4] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[2,4,5,1,3] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[2,5,4,1,3] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[4,1,3,5,2] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition

$\lambda \vdash n$ by the number of standard tableaux

$T$ of shape

$\lambda$ such that

$\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].

Matching statistic: St000668
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%

Values

[1,2] => [1] => [1]
 => []
 => ? ∊ {1,2}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2

Description

The least common multiple of the parts of the partition.

Matching statistic: St000707
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%

Values

[1,2] => [1] => [1]
 => []
 => ? ∊ {1,2}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2

Description

The product of the factorials of the parts.

Matching statistic: St000708
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%

Values

[1,2] => [1] => [1]
 => []
 => ? ∊ {1,2}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2

Description

The product of the parts of an integer partition.

Matching statistic: St000933
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%

Values

[1,2] => [1] => [1]
 => []
 => ? ∊ {1,2}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2

Description

The number of multipartitions of sizes given by an integer partition. This is, for

$\lambda = (\lambda_1,\ldots,\lambda_n)$ , this is the number of

$n$ -tuples

$(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions

$\lambda^{(i)}$ such that

$\lambda^{(i)} \vdash \lambda_i$ .

Matching statistic: St000937
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 52%●distinct values known / distinct values provided: 43%

Values

[1,2] => [1] => [1]
 => []
 => ? ∊ {1,2}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2

Description

The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation

$S^{(2,2)}$ are

$2$ on the conjugacy classes

$(4)$ and

$(2,2)$ ,

$0$ on the conjugacy classes

$(3,1)$ and

$(1,1,1,1)$ , and

$-1$ on the conjugacy class

$(2,1,1)$ . Therefore, the statistic on the partition

$(2,2)$ is

$2$ .

Matching statistic: St001878
(load all 4 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 49%●distinct values known / distinct values provided: 29%

Values

[1,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,2,3] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[1,3,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[2,1,3] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[2,3,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[3,1,2] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[3,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,2,2,2,3}
[1,2,3,4] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2,4,5] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,4,2,5] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,4,5,2,3] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,3,1,5,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,3,2,1,5] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,3,2,5,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,1,4,3,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,2,1,4] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,4,2,1,3] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,4,3,1,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,5,4,6] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,5,6,4] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,6,4,5] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,4,3,6,5] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,6,4] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,5,4,3,6] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,5,4,6,3] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,6,4,3,5] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,4,6,5] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4,6] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2,5,6,4] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2,6,4,5] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2,6,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,3,4,2,6,5] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,5,2,6,4] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,2,6] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,6,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,6,4,2,5] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,6,5,2,4] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,6,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,4,2,3,6,5] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3,6] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1

Description

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree.