Your data matches 215 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001705
Mapping
St001705: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of occurrences of the pattern 2413 in a permutation.
Matching statistic: St000687
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
 => []
 => []
 => ? = 0
[1,2] => [1,0,1,0]
 => [1]
 => [1,0,1,0]
 => 0
[2,1] => [1,1,0,0]
 => []
 => []
 => ? = 0
[1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,3] => [1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,3,1] => [1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,2] => [1,1,1,0,0,0]
 => []
 => []
 => ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
 => []
 => []
 => ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St000205
(load all 10 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
 => []
 => ? = 0
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 0
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 0
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0}
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1}
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2}
[3,2,1,4,5,6] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[3,2,1,4,6,5] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[3,2,1,5,4,6] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[3,2,1,5,6,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[3,2,1,6,4,5] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[3,2,1,6,5,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St001541
(load all 3 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1]
 => []
 => ? = 0
[1,2] => [1,2] => [1,1]
 => [1]
 => 0
[2,1] => [2,1] => [2]
 => []
 => ? = 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,3,1] => [3,2,1] => [3]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [3,2,1] => [3]
 => []
 => ? ∊ {0,0,0}
[3,2,1] => [3,2,1] => [3]
 => []
 => ? ∊ {0,0,0}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [1,4,3,2] => [3,1]
 => [1]
 => 0
[1,4,2,3] => [1,4,3,2] => [3,1]
 => [1]
 => 0
[1,4,3,2] => [1,4,3,2] => [3,1]
 => [1]
 => 0
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1
[2,3,1,4] => [3,2,1,4] => [3,1]
 => [1]
 => 0
[2,3,4,1] => [4,2,3,1] => [3,1]
 => [1]
 => 0
[2,4,1,3] => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 0
[2,4,3,1] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[3,1,2,4] => [3,2,1,4] => [3,1]
 => [1]
 => 0
[3,1,4,2] => [4,2,3,1] => [3,1]
 => [1]
 => 0
[3,2,1,4] => [3,2,1,4] => [3,1]
 => [1]
 => 0
[3,2,4,1] => [4,2,3,1] => [3,1]
 => [1]
 => 0
[3,4,1,2] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[3,4,2,1] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[4,1,2,3] => [4,2,3,1] => [3,1]
 => [1]
 => 0
[4,1,3,2] => [4,2,3,1] => [3,1]
 => [1]
 => 0
[4,2,1,3] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[4,2,3,1] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[4,3,1,2] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[4,3,2,1] => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,3,5,2,4] => [1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,5,2,3,4] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,5,2,4,3] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
 => [1]
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
 => [2]
 => 1
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
 => [2]
 => 1
[2,4,5,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[2,5,3,4,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[2,5,4,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,4,5,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,5,1,4,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,5,2,4,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,5,4,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[3,5,4,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,2,5,1,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,2,5,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,3,5,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,1,3,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,2,1,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,2,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,3,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[4,5,3,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,2,3,1,4] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,2,3,4,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,2,4,1,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,1,4,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2}
[2,4,6,5,3,1] => [6,5,4,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,5,6,3,4,1] => [6,5,4,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,5,6,4,3,1] => [6,5,4,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St001912
(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
St001912: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Values
[1] => [] => []
 => ?
 => ? = 0
[1,2] => [1] => [1]
 => []
 => ? ∊ {0,0}
[2,1] => [1] => [1]
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => 0
[1,3,2] => [1,2] => [1,1]
 => [1]
 => 0
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {0,0,0}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => 0
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0,0}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => 0
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => 0
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[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]
 => 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[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]
 => 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => 0
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => 0
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => 0
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[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]
 => 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => 0
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => 0
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => 0
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,5,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,4,2,5] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,4,5,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,5,4,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,2,1,5] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,2,5,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,5,2,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,5,1,4,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[3,5,4,2,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,2,3,5] => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,2,5,3] => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,5,2,3] => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,1,2,5] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,1,5,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,5,1,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,5,1,2,3] => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[4,5,3,1,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,2,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,2,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,3,1,4,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,3,4,2,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,4,1,2,3] => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[5,4,3,1,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
[2,3,4,5,1,6] => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,3,4,5,6,1] => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,3,4,6,5,1] => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,3,5,1,4,6] => [2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,3,5,1,6,4] => [2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[2,3,5,6,1,4] => [2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
Matching statistic: St000934
(load all 6 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 75%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
 => []
 => ? = 0
[1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0}
[2,1,3] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0}
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[3,1,2] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[3,2,1] => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[3,1,2,4] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1}
[3,2,4,1] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,3,2] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[4,2,1,3] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[1,4,5,2,3] => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 0
[1,4,5,3,2] => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 0
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[2,1,4,5,3] => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 0
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 0
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,3,5,1,4] => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3,5] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[2,4,3,5,1] => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[2,4,5,1,3] => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[2,4,5,3,1] => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[2,5,1,3,4] => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[2,5,1,4,3] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[2,5,4,1,3] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[2,5,4,3,1] => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[3,1,4,2,5] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,5,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,2,1,5] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,3,5,2] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,2,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,1,3,5] => [1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,1,5,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,3,1,5] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1,5] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5}
Description
The 2-degree of an integer partition. For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
Matching statistic: St000936
(load all 11 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
 => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [2]
 => []
 => ? ∊ {0,0}
[2,1] => [2]
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0}
[1,3,2] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,2,1] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [2,2]
 => [2]
 => 0
[1,2,4,3] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,3,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,2,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,4,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,2] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,3,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,3,1] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,2,1] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [2,2,1]
 => [2,1]
 => 2
[1,2,3,5,4] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,3,4,5,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,5,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,5,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,5,4,2,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,3,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,4,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,1,4,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,3,1,5,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,3,4,1,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,3,4,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,3,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,4,3,5,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,5,1,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,4,5,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,3,1,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,4,1,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,2,1,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,4,1,2,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,5,1,4,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,2,5,1,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,3,2,1,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,5,3,1,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,2,4,3,1] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
Description
The number of even 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 conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
(load all 9 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 73%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
 => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [2]
 => []
 => ? ∊ {0,0}
[2,1] => [2]
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0}
[1,3,2] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,2,1] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [2,2]
 => [2]
 => 0
[1,2,4,3] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,3,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,2,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,4,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,2] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,3,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,3,1] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,2,1] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [2,2,1]
 => [2,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,3,4,5,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,5,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,5,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,5,4,2,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,3,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,4,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,3,1,5,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,3,4,1,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,3,4,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,3,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,4,3,5,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,5,1,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,4,5,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,3,1,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,4,1,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,1,4,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,5,1,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,3,1,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of zeros 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: St000940
(load all 8 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
 => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [2]
 => []
 => ? ∊ {0,0}
[2,1] => [2]
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0}
[1,3,2] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,2,1] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [2,2]
 => [2]
 => 0
[1,2,4,3] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,3,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,2,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,4,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,2] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,3,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,3,1] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,2,1] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [2,2,1]
 => [2,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,3,4,5,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,5,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,5,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,5,4,2,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,3,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,4,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,3,1,5,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,3,4,1,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,3,4,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,3,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,4,3,5,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,5,1,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,4,5,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,3,1,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,4,1,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,1,4,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,5,1,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,3,1,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2].
Matching statistic: St001124
(load all 10 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
 => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [2]
 => []
 => ? ∊ {0,0}
[2,1] => [2]
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0}
[1,3,2] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,1,2] => [3]
 => [3]
 => []
 => ? ∊ {0,0,0}
[3,2,1] => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [2,2]
 => [2]
 => 0
[1,2,4,3] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,4,3,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,2,4] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,4] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,4,1] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,2] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,1,3,2] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,3] => [3,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[4,2,3,1] => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,2,1] => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [2,2,1]
 => [2,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,3,4,5,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,2,5,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,5,2] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,5,4,2,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,3,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,5,4,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,3,1,5,4] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,3,4,1,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,3,4,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,3,5] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,1,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
 => [3,2]
 => [2]
 => 0
[2,4,3,5,1] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,4,5,1,3] => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[2,4,5,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,3,1,4] => [4,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,5,4,1,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,1,4,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,5,1,3] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1,5] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,5,3,1,2] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [2,2,1]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 205 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000379The number of Hamiltonian cycles in a graph. St000929The constant term of the character polynomial of an integer partition. St001651The Frankl number of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001095The number of non-isomorphic posets with precisely one further covering relation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St000323The minimal crossing number of a graph. St000370The genus of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St000447The number of pairs of vertices of a graph with distance 3. St001793The difference between the clique number and the chromatic number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001586The number of odd parts smaller than the largest even part in an integer partition. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000699The toughness times the least common multiple of 1,. St000567The sum of the products of all pairs of parts. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000781The number of proper colouring schemes of a Ferrers diagram. St000322The skewness of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001578The minimal number of edges to add or remove to make a graph a line graph. St000095The number of triangles of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000068The number of minimal elements in a poset. St001330The hat guessing number of a graph. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000475The number of parts equal to 1 in a partition. St000842The breadth of a permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001568The smallest positive integer that does not appear twice in the partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000069The number of maximal elements of a poset. St001621The number of atoms of a lattice. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000629The defect of a binary word. St001868The number of alignments of type NE of a signed permutation. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001867The number of alignments of type EN of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000449The number of pairs of vertices of a graph with distance 4. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000552The number of cut vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001281The normalized isoperimetric number of a graph. St001549The number of restricted non-inversions between exceedances. St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St000264The girth of a graph, which is not a tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.