Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001183
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[3,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[5,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[6,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[7,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001258
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[3,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[5,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[6,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[7,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St001526
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[3,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[4,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[5,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[6,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[7,1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 2
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000619
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[2,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[3,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[4,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[5,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 1
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
[6,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 1
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 3 - 1
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2 = 3 - 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
[7,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 1
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1 = 2 - 1
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 3 - 1
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3 = 4 - 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2 = 3 - 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2 = 3 - 1
[8,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[7,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 2 - 1
[7,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[6,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 1
[9,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[10,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[11,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[12,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[13,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[14,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[15,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
[16,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 1
Description
The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St001499
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[2,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[3,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[4,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[5,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[6,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[7,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[8,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[7,2]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[7,1,1]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[6,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[9,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[10,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[11,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[12,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[13,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[14,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[15,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[16,1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001960
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[2,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[3,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[2,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 2 - 2
[4,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[3,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 3 - 2
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 2 - 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 2 - 2
[5,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[4,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 3 - 2
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 2 - 2
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 3 - 2
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 3 - 2
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 2 - 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 2 - 2
[6,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[5,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 3 - 2
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 2 - 2
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 3 - 2
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 3 - 2
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 3 - 2
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 2 - 2
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 3 - 2
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1 = 3 - 2
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 2 - 2
[7,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[6,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 3 - 2
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 2 - 2
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 2 - 2
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 3 - 2
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 3 - 2
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 3 - 2
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 2 - 2
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 4 - 2
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1 = 3 - 2
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 3 - 2
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1 = 3 - 2
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 2 - 2
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2 = 4 - 2
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1 = 3 - 2
[8,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[7,2]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[7,1,1]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0 = 2 - 2
[6,3]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[9,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[10,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[11,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[12,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[13,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[14,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[15,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
[16,1]
 => [1]
 => [1,0]
 => [1] => ? = 1 - 2
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000075
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 60%
Values
[1,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[2,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[3,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[2,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[4,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[3,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[5,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[4,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => ? = 2
[6,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[5,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 3
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => ? = 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => ? = 2
[7,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[6,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => ? = 2
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 3
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 4
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 3
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => ? = 3
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 3
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => ? = 2
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => ? = 3
[8,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[7,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[7,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[6,3]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2
[6,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[6,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[5,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => ? = 2
[5,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 3
[5,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 3
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[4,4,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 3
[4,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 4
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 3
[4,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => ? = 3
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 3
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => ? = 2
[3,3,3]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => ? = 3
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 4
[3,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => ? = 3
[2,2,2,2,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[1,2,3,4,8],[5,6,7,9,10]]
 => ? = 4
[9,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[8,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[8,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
[7,3]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2
[7,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3
[7,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 2
[6,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => ? = 2
[6,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => ? = 3
[6,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3
[6,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 3
[6,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => ? = 2
[5,5]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => ? = 2
[5,4,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 3
[5,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 4
[5,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 3
[5,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => ? = 3
[5,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 3
[5,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => ? = 2
[4,4,2]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 4
[4,3,3]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => ? = 3
[4,3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 4
[10,1]
 => [1]
 => [1,0]
 => [[1],[2]]
 => 1
[9,2]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[9,1,1]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 2
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St001582
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 60%
Values
[1,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[2,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[4,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2
[5,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 2
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 3
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 2
[6,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 2
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 3
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 3
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 3
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 3
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 2
[7,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 2
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 2
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 3
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 3
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 4
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 3
[3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 3
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 3
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 2
[2,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 4
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 3
[8,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[7,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[6,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 2
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 2
[5,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 2
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 3
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 3
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2
[4,4,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 3
[4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 4
[4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 3
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 3
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 3
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 2
[3,3,3]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 3
[3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? = 4
[3,2,2,2]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 4
[3,2,2,1,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 3
[2,2,2,2,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 4
[9,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[8,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[10,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[9,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[9,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[8,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[11,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[10,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[10,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[9,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[12,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[11,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[11,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[10,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[13,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[12,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[12,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[11,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[14,1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1
[13,2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[13,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[12,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.