Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1,0]
 => [1,0]
 => 0
[1,2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2
[3,4,1,2] => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2
[4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,4,5,2,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,5,4,3,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,1,3,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,1,4,3,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,5,3,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,4,5,1,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,5,4,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,1,2,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,1,2,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,1,5,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,5,2,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,5,1,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,5,1,4,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,5,4,1,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,1,5,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,2,5,1,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[4,3,2,1,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[4,3,2,5,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,5,1,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,5,1,3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,5,2,1,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,5,3,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[5,1,4,3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[5,2,4,3,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[5,3,2,1,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[5,3,2,4,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[5,3,4,2,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[5,4,1,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[5,4,2,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[5,4,3,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,3,2,5,6,4] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,3,2,6,4,5] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,3,4,2,6,5] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,3,5,6,2,4] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001200
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1] => [1]
 => [1]
 => [1,0]
 => ? = 0 + 1
[1,2] => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1] => [2]
 => [1,1]
 => [1,1,0,0]
 => ? = 0 + 1
[1,3,2] => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[3,1,2] => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[3,2,1] => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [2,2]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [2,2]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [2,2]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,5,4,3,2] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,3,5,4] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,4,3,5] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,4,5,3] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,5,3,4] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,5,4,3] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,3,1,5,4] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,4,5,1,3] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,5,4,3,1] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,1,2,5,4] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,2,1,5,4] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,4,1,2,5] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,4,1,5,2] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,4,5,2,1] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,5,1,2,4] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,5,1,4,2] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,5,4,1,2] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,1,5,2,3] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,2,5,1,3] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[4,3,2,1,5] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[4,3,2,5,1] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,3,5,1,2] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,5,1,3,2] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,5,2,1,3] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,5,3,1,2] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[5,1,4,3,2] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[5,2,4,3,1] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[5,3,2,1,4] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[5,3,2,4,1] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[5,3,4,2,1] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[5,4,1,2,3] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[5,4,2,3,1] => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[5,4,3,2,1] => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,3,2,5,6,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,2,6,4,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,4,2,6,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,5,6,2,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,6,5,4,2] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,4,2,3,6,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,4,5,2,6,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,4,5,6,3,2] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,4,6,2,3,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,4,6,5,2,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,5,2,6,3,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,5,4,3,6,2] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,5,4,6,2,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,5,6,2,4,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,5,6,3,2,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,6,2,5,4,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,6,4,3,2,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,6,4,5,3,2] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,6,5,2,3,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,6,5,3,4,2] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,3,5,6,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,3,6,4,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,4,3,6,5] => [2,2,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,4,5,3,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,4,6,5,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,5,3,4,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,5,4,6,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,5,6,3,4] => [2,2,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,6,3,5,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,6,4,3,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,6,5,4,3] => [2,2,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,3,1,4,6,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,3,1,5,4,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,3,1,6,5,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,4,3,1,6,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,4,5,1,3,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,4,6,1,5,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,5,3,6,1,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,5,4,3,1,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,5,6,4,1,3] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,6,3,5,4,1] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,6,4,3,5,1] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,6,5,4,3,1] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,1,2,4,6,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,1,2,5,4,6] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,1,2,6,5,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,2,1,5,6,4] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,2,1,6,4,5] => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 3 + 1
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000510
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [-1] => [1]
 => ? = 0
[1,2] => [1,2] => [-1,-2] => [1,1]
 => 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 2
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 2
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 2
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 2
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 2
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 2
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 2
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 2
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 2
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 2
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 2
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 2
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 2
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 2
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 2
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 2
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 2
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 2
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 2
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 2
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3
[2,1,5,4,6,3] => [2,1,5,4,6,3] => [-2,-1,-5,-4,-6,-3] => ?
 => ? = 3
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000681
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [-1] => [1]
 => ? = 0
[1,2] => [1,2] => [-1,-2] => [1,1]
 => 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 2
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 2
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 2
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 2
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 2
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 2
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 2
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 2
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 2
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 2
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 2
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 2
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 2
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 2
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 2
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 2
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 2
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 2
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 2
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 2
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3
[2,1,5,4,6,3] => [2,1,5,4,6,3] => [-2,-1,-5,-4,-6,-3] => ?
 => ? = 3
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000512
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 1
[1,2] => [1,2] => [-1,-2] => [1,1]
 => 0 = 1 - 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 1 = 2 - 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 1 = 2 - 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 1
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 1
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 1 = 2 - 1
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 1
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 1
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 1
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 1
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 1
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 1
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 1
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 1
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 1
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 1
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 1
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 1
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 1
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 1
[2,1,5,4,6,3] => [2,1,5,4,6,3] => [-2,-1,-5,-4,-6,-3] => ?
 => ? = 3 - 1
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000934
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 1
[1,2] => [1,2] => [-1,-2] => [1,1]
 => 0 = 1 - 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 1 = 2 - 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 1 = 2 - 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 1
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 1
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 1 = 2 - 1
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 1
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 1
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 1
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 1
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 1
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 1
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 1
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 1
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 1
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 1
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 1
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 1
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 1
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 1
[2,1,5,4,6,3] => [2,1,5,4,6,3] => [-2,-1,-5,-4,-6,-3] => ?
 => ? = 3 - 1
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: St000997
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000997: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 3
[1,2] => [1,2] => [-1,-2] => [1,1]
 => -2 = 1 - 3
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 3
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 3
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 3
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => -1 = 2 - 3
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => -1 = 2 - 3
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 3
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 3
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 3
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 3
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 3
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 3
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 3
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => -1 = 2 - 3
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => -1 = 2 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 3
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => -1 = 2 - 3
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => -1 = 2 - 3
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => -1 = 2 - 3
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => -1 = 2 - 3
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 3
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 3
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => -1 = 2 - 3
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => -1 = 2 - 3
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => -1 = 2 - 3
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 3
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => -1 = 2 - 3
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => -1 = 2 - 3
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 3
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 3
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => -1 = 2 - 3
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => -1 = 2 - 3
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => -1 = 2 - 3
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => -1 = 2 - 3
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 3
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => -1 = 2 - 3
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 3
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => -1 = 2 - 3
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 3
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => -1 = 2 - 3
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => -1 = 2 - 3
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => -1 = 2 - 3
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 3
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 3
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 3
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 3
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 3
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 3
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 3
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 3
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 3
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 3
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 3
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 3
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 3
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 3
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 3
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 3
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 3
[2,1,5,4,6,3] => [2,1,5,4,6,3] => [-2,-1,-5,-4,-6,-3] => ?
 => ? = 3 - 3
Description
The even-odd crank of an integer partition. This is the largest even part minus the number of odd parts.
Matching statistic: St001603
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 1
[1,2] => [1,2] => [-1,-2] => [1,1]
 => ? = 1 - 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 1 = 2 - 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 1 = 2 - 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 1
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 1
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 1 = 2 - 1
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 1
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 1
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 1
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 1
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 1
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 1
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 1
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 1
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 1
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 1
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 1
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 1
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 1
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 1
[1,2] => [1,2] => [-1,-2] => [1,1]
 => ? = 1 - 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 1 = 2 - 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 1 = 2 - 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 1
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 1
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 1 = 2 - 1
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 1
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 1
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 1
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 1
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 1
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 1
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 1
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 1
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 1
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 1
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 1
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 1
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 1
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [-1] => [1]
 => ? = 0 - 1
[1,2] => [1,2] => [-1,-2] => [1,1]
 => ? = 1 - 1
[2,1] => [2,1] => [-2,-1] => []
 => ? = 0 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [1]
 => ? = 1 - 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => [3]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => ? = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => ? = 3 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
 => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
 => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
 => 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
 => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
 => 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
 => ? = 3 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
 => 1 = 2 - 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
 => 1 = 2 - 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
 => 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
 => ? = 3 - 1
[5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => [3]
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
 => ? = 3 - 1
[5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => [3]
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
 => ? = 3 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => [3]
 => 1 = 2 - 1
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [3]
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => [3]
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
 => ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 3 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 3 - 1
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 3 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [-1,-4,-6,-2,-3,-5] => ?
 => ? = 3 - 1
[1,4,6,5,2,3] => [1,4,6,5,2,3] => ? => ?
 => ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [-1,-5,-2,-6,-3,-4] => ?
 => ? = 3 - 1
[1,5,4,3,6,2] => [1,5,4,3,6,2] => [-1,-5,-4,-3,-6,-2] => ?
 => ? = 3 - 1
[1,5,4,6,2,3] => [1,5,4,6,2,3] => ? => ?
 => ? = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ? => ?
 => ? = 3 - 1
[1,5,6,3,2,4] => [1,5,6,3,2,4] => ? => ?
 => ? = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,4,3] => ? => ?
 => ? = 3 - 1
[1,6,4,3,2,5] => [1,6,4,3,2,5] => ? => ?
 => ? = 3 - 1
[1,6,4,5,3,2] => [1,6,4,5,3,2] => [-1,-6,-4,-5,-3,-2] => ?
 => ? = 3 - 1
[1,6,5,2,3,4] => [1,6,5,2,3,4] => ? => ?
 => ? = 3 - 1
[1,6,5,3,4,2] => [1,6,5,3,4,2] => ? => ?
 => ? = 3 - 1
[2,1,3,5,6,4] => [2,1,3,5,6,4] => [-2,-1,-3,-5,-6,-4] => ?
 => ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,4,5] => [-2,-1,-3,-6,-4,-5] => ?
 => ? = 3 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [-2,-1,-4,-3,-6,-5] => ?
 => ? = 3 - 1
[2,1,4,5,3,6] => [2,1,4,5,3,6] => [-2,-1,-4,-5,-3,-6] => ?
 => ? = 3 - 1
[2,1,4,6,5,3] => [2,1,4,6,5,3] => [-2,-1,-4,-6,-5,-3] => ?
 => ? = 3 - 1
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [-2,-1,-5,-3,-4,-6] => ?
 => ? = 3 - 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.