- FindStat

Your data matches 37 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000708

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The product of the parts of an integer partition.

Matching statistic: St001786

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,4,1,2] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,5,2,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,5,4,3,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,5,3,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[2,4,5,1,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,5,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[2,5,4,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,1,2,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,2,1,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,2,5,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[3,4,1,5,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,4,5,2,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 12
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[3,5,7,9,10,12,1,2,4,6,8,11] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[3,5,8,9,11,12,1,2,4,6,7,10] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[3,6,7,8,10,12,1,2,4,5,9,11] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[3,6,7,9,10,12,1,2,4,5,8,11] => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 12
[3,6,7,9,11,12,1,2,4,5,8,10] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[3,6,8,9,10,12,1,2,4,5,7,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[3,7,8,9,10,12,1,2,4,5,6,11] => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 12
[4,7,8,9,10,12,1,2,3,5,6,11] => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 6
[4,8,9,10,11,12,1,2,3,5,6,7] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[5,7,9,10,11,12,1,2,3,4,6,8] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[5,8,9,10,11,12,1,2,3,4,6,7] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[6,7,8,10,11,12,1,2,3,4,5,9] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[6,7,9,10,11,12,1,2,3,4,5,8] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,4,3,6,5,9,11,8,12,10,7] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,4,3,7,9,6,10,8,5,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,4,3,7,9,6,11,8,12,10,5] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,4,3,7,10,6,11,12,9,8,5] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,4,3,8,9,11,7,6,12,10,5] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,4,3,8,10,11,7,12,9,6,5] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,5,7,4,8,6,3,10,9,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,5,7,4,9,6,10,8,3,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,5,7,4,10,6,11,12,9,8,3] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,5,8,4,9,10,7,6,3,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,5,8,4,9,11,7,6,12,10,3] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,6,7,9,5,4,10,8,3,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,6,7,9,5,4,11,8,12,10,3] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,6,7,10,5,4,11,12,9,8,3] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[2,1,6,8,9,5,10,7,4,3,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,7,9,10,11,6,12,8,5,4,3] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[3,5,2,6,4,1,8,7,10,9,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[3,5,2,7,4,8,6,1,10,9,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[3,5,2,8,4,9,10,7,6,1,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[3,6,2,7,8,5,4,1,10,9,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[3,6,2,7,9,5,4,10,8,1,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[3,6,2,7,10,5,4,11,12,9,8,1] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[4,5,7,3,2,8,6,1,10,9,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[4,5,7,3,2,9,6,10,8,1,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[4,5,7,3,2,10,6,11,12,9,8,1] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[4,5,8,3,2,9,10,7,6,1,12,11] => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[4,5,8,3,2,9,11,7,6,12,10,1] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4
[4,6,7,3,8,5,2,1,10,9,12,11] => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 4

Description

The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. Alternatively, remark that the monomials of the polynomial

$\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with

$n$ east steps: the exponent of

$z_i$ is the number of north steps before the

$i$ -th east step, see [2]. Thus, this statistic records the coefficients of the monomials. A formula for the coefficient of

$z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]:

$c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.$ This polynomial arises in a partial symmetrization process as follows, see [1]. For

$w\in\frak{S}_n$ , let

$w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$ . Furthermore, let

$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$ Then

$\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$ .

Matching statistic: St000609
(load all 2 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 27% ●values known / values provided: 29%●distinct values known / distinct values provided: 27%

Values

[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[3,2,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[4,2,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[1,5,2,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,5,4,3,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[2,1,3,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[2,1,3,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[2,1,4,3,5] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[2,1,4,5,3] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[2,1,5,3,4] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[2,1,5,4,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[2,3,1,4,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[2,3,1,5,4] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[2,4,3,1,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[2,4,5,1,3] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[2,5,3,4,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[2,5,4,3,1] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[3,1,2,4,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[3,1,2,5,4] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[3,2,1,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[3,2,4,1,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[3,2,5,4,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
[3,4,1,2,5] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
[3,4,1,5,2] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[3,4,5,2,1] => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,6,5,4,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[8,6,7,5,4,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[6,7,8,5,4,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[8,7,5,6,4,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[7,8,5,6,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,5,6,7,4,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[5,6,7,8,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => {{1,4},{2,6},{3,8},{5},{7}}
 => ? = 4 - 1
[7,8,6,4,5,3,2,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => ? = 2 - 1
[8,6,7,4,5,3,2,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => ? = 2 - 1
[8,7,5,4,6,3,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[8,7,4,5,6,3,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[8,6,5,4,7,3,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[8,5,6,4,7,3,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[6,7,5,4,8,3,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[7,5,6,4,8,3,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[5,6,7,4,8,3,2,1] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
[7,5,4,6,8,3,2,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 3 - 1
[5,6,4,7,8,3,2,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 3 - 1
[6,4,5,7,8,3,2,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 3 - 1
[5,4,6,7,8,3,2,1] => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => ? = 6 - 1
[8,7,6,5,3,4,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[7,8,6,5,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,6,3,4,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,5,6,3,4,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[6,5,7,8,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,3,5,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[7,8,6,4,3,5,2,1] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
[8,6,7,4,3,5,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[6,7,8,4,3,5,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[8,7,6,3,4,5,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[7,8,6,3,4,5,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => {{1,4},{2,6},{3,8},{5},{7}}
 => ? = 4 - 1
[7,8,5,4,3,6,2,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => ? = 2 - 1
[8,7,4,5,3,6,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[7,8,5,3,4,6,2,1] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => {{1,4},{2,6},{3,8},{5},{7}}
 => ? = 4 - 1
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => {{1,6},{2,8},{3},{4},{5},{7}}
 => ? = 2 - 1
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 1 - 1
[8,6,5,4,3,7,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[8,6,4,3,5,7,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[8,5,4,3,6,7,2,1] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
[8,5,3,4,6,7,2,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => ? = 2 - 1
[8,4,3,5,6,7,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[7,6,5,4,3,8,2,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 2 - 1
[6,7,5,4,3,8,2,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 4 - 1
[6,5,7,4,3,8,2,1] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
[6,7,4,5,3,8,2,1] => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => ? = 6 - 1
[6,5,4,7,3,8,2,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 3 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

Matching statistic: St000589

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 27%

Values

[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[1,5,2,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,5,4,3,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,3,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[2,1,3,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,4,3,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,4,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,1,5,3,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,1,5,4,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,3,1,4,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,3,1,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,4,3,1,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,4,5,1,3] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,5,3,4,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,5,4,3,1] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,1,2,4,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,1,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[3,2,1,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[3,2,4,1,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,2,5,4,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,4,1,2,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[3,4,1,5,2] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,4,5,2,1] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,6,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,6,7,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[6,7,8,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,7,5,6,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,5,6,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,5,6,7,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[5,6,7,8,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[7,8,6,4,5,3,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,6,7,4,5,3,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[7,6,8,4,5,3,2,1] => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
[8,7,5,4,6,3,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,7,4,5,6,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,6,5,4,7,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[8,5,6,4,7,3,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[6,7,5,4,8,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[7,5,6,4,8,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[5,6,7,4,8,3,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[7,5,4,6,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[5,6,4,7,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[6,4,5,7,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[5,4,6,7,8,3,2,1] => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 6 - 1
[8,7,6,5,3,4,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,6,5,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,6,3,4,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,5,6,3,4,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[6,5,7,8,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,3,5,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[7,8,6,4,3,5,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[8,6,7,4,3,5,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[6,7,8,4,3,5,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[8,7,6,3,4,5,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,6,3,4,5,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[7,8,5,4,3,6,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,7,4,5,3,6,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[7,8,5,3,4,6,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2 - 1
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
[8,6,5,4,3,7,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,6,4,3,5,7,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,5,4,3,6,7,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,5,3,4,6,7,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,4,3,5,6,7,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[7,6,5,4,3,8,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[6,7,5,4,3,8,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[6,5,7,4,3,8,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[6,7,4,5,3,8,2,1] => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 6 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.

Matching statistic: St000612

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 27%

Values

[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[1,5,2,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,5,4,3,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,3,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[2,1,3,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,4,3,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,1,4,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,1,5,3,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,1,5,4,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[2,3,1,4,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,3,1,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,4,3,1,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,4,5,1,3] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[2,5,3,4,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[2,5,4,3,1] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,1,2,4,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,1,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[3,2,1,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[3,2,4,1,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,2,5,4,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[3,4,1,2,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[3,4,1,5,2] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[3,4,5,2,1] => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,6,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,6,7,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[6,7,8,5,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,7,5,6,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,5,6,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,5,6,7,4,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[5,6,7,8,4,3,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[7,8,6,4,5,3,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,6,7,4,5,3,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[7,6,8,4,5,3,2,1] => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
[8,7,5,4,6,3,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,7,4,5,6,3,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,6,5,4,7,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[8,5,6,4,7,3,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[6,7,5,4,8,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[7,5,6,4,8,3,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[5,6,7,4,8,3,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[7,5,4,6,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[5,6,4,7,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[6,4,5,7,8,3,2,1] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
[5,4,6,7,8,3,2,1] => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 6 - 1
[8,7,6,5,3,4,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,6,5,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,6,3,4,2,1] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
[7,8,5,6,3,4,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[6,5,7,8,3,4,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,6,4,3,5,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[7,8,6,4,3,5,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[8,6,7,4,3,5,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[6,7,8,4,3,5,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[8,7,6,3,4,5,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[7,8,6,3,4,5,2,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4 - 1
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[7,8,5,4,3,6,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,7,4,5,3,6,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[7,8,5,3,4,6,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 4 - 1
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2 - 1
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
[8,6,5,4,3,7,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,6,4,3,5,7,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[8,5,4,3,6,7,2,1] => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
[8,5,3,4,6,7,2,1] => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
[8,4,3,5,6,7,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[7,6,5,4,3,8,2,1] => [5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 2 - 1
[6,7,5,4,3,8,2,1] => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 4 - 1
[6,5,7,4,3,8,2,1] => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
[6,7,4,5,3,8,2,1] => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 6 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.

Matching statistic: St001232

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[3,4,1,2] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,2,4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,5,3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,3,4,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,4,2,3,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,4,5,2,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,5,2,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,5,3,2,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,5,4,3,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[2,1,3,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,1,4,3,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,5,3,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,3,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[2,4,5,1,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,5,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[2,5,4,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[3,1,2,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,2,1,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[3,2,1,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[3,2,4,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[3,2,5,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[3,4,1,2,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[3,4,1,5,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,4,5,2,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,5,1,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,5,1,4,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[3,5,4,1,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,1,3,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[4,1,5,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,2,1,3,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[4,2,3,1,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[4,2,3,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[4,2,5,1,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[4,3,2,1,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[4,3,2,5,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,3,5,1,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,5,1,3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,5,2,1,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,5,3,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[5,1,3,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[5,1,4,3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[5,2,1,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[5,2,3,1,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[5,3,2,1,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[5,3,4,2,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[5,4,1,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[5,4,2,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,4,5,6,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,4,6,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,5,3,6,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,5,6,4,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,6,3,4,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,6,5,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,3,1,5,6,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,6,4,5] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,3,4,1,6,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,3,5,6,1,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,3,6,5,4,1] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,4,1,3,6,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,4,5,1,6,3] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,4,5,6,3,1] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,4,6,1,3,5] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,4,6,5,1,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,5,1,6,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,5,4,3,6,1] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,5,4,6,1,3] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,5,6,1,4,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,5,6,3,1,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,6,1,5,4,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,6,4,3,1,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,6,4,5,3,1] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,6,5,1,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,6,5,3,4,1] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[3,1,2,5,6,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 7%

Values

[1,2,3] => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => ? = 1 - 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,5,2,4,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,5,3,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,5,3,4,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,5,4,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,4,3,1,5] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,5,3,4,1] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,2,1,5,4] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,2,5,4,1] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,5,1,2,4] => [3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,5,1,4,2] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,5,4,1,2] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,1,3,2,5] => [3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4,1,5,2,3] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,2,1,3,5] => [2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4,2,3,1,5] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4,2,3,5,1] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4,2,5,1,3] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,3,2,1,5] => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,3,2,5,1] => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,3,5,1,2] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,5,1,3,2] => [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,5,2,1,3] => [4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[4,5,3,1,2] => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,1,3,4,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[5,1,4,3,2] => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,2,1,4,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[5,2,3,1,4] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[5,2,3,4,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[5,2,4,3,1] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,3,2,1,4] => [3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,3,2,4,1] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,3,4,2,1] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,4,1,2,3] => [4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,4,2,3,1] => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[5,4,3,2,1] => [3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
 => ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001593

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 13%

Values

[1,2,3] => [1,2,3] => [-1,-2,-3] => [1,1,1]
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [1,1]
 => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [1,1]
 => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [1,1]
 => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [1,1]
 => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
 => ? = 2 - 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [1,1]
 => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [1,1]
 => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
 => ? = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => [3,1,1]
 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => [3,1,1]
 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [1,1,1]
 => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [1,1,1]
 => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => [3,1,1]
 => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => [3,1,1]
 => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => [3,1,1]
 => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [1,1,1]
 => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => [3,1,1]
 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
 => 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => [3,1,1]
 => 0 = 1 - 1
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => [3,1,1]
 => 0 = 1 - 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [1,1,1]
 => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
 => 1 = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [1,1,1]
 => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
 => 1 = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
 => 1 = 2 - 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]
 => 1 = 2 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => [3,1,1]
 => 0 = 1 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
 => 1 = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => [3,1,1]
 => 0 = 1 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
 => 1 = 2 - 1
[2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => [3,1,1]
 => 0 = 1 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
 => 1 = 2 - 1
[3,1,2,4,5] => [3,1,2,4,5] => [-3,-1,-2,-4,-5] => [3,1,1]
 => 0 = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
 => 1 = 2 - 1
[3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => [1,1,1]
 => 0 = 1 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
 => 1 = 2 - 1
[3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => [3,1,1]
 => 0 = 1 - 1
[3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => [3,1,1]
 => 0 = 1 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
 => 1 = 2 - 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]
 => 1 = 2 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
 => 1 = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [-1,-2,-3,-4,-5,-6] => ?
 => ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [-1,-2,-3,-4,-6,-5] => ?
 => ? = 1 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [-1,-2,-3,-5,-4,-6] => ?
 => ? = 1 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [-1,-2,-3,-5,-6,-4] => ?
 => ? = 1 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [-1,-2,-3,-6,-4,-5] => ?
 => ? = 1 - 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [-1,-2,-3,-6,-5,-4] => ?
 => ? = 1 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [-1,-2,-4,-3,-5,-6] => ?
 => ? = 1 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [-1,-2,-4,-3,-6,-5] => ?
 => ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [-1,-2,-4,-5,-3,-6] => ?
 => ? = 1 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [-1,-2,-4,-5,-6,-3] => ?
 => ? = 1 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [-1,-2,-4,-6,-3,-5] => ?
 => ? = 1 - 1
[1,2,4,6,5,3] => [1,2,4,6,5,3] => [-1,-2,-4,-6,-5,-3] => ?
 => ? = 1 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [-1,-2,-5,-3,-4,-6] => ?
 => ? = 1 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [-1,-2,-5,-3,-6,-4] => ?
 => ? = 1 - 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [-1,-2,-5,-4,-3,-6] => ?
 => ? = 1 - 1
[1,2,5,4,6,3] => [1,2,5,4,6,3] => [-1,-2,-5,-4,-6,-3] => ?
 => ? = 1 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [-1,-2,-5,-6,-3,-4] => ?
 => ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,4,3] => [-1,-2,-5,-6,-4,-3] => ?
 => ? = 1 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [-1,-2,-6,-3,-4,-5] => ?
 => ? = 1 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ? => ?
 => ? = 1 - 1
[1,2,6,4,3,5] => [1,2,6,4,3,5] => ? => ?
 => ? = 1 - 1
[1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ?
 => ? = 1 - 1
[1,2,6,5,3,4] => [1,2,6,5,3,4] => ? => ?
 => ? = 1 - 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ?
 => ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [-1,-3,-2,-4,-5,-6] => ?
 => ? = 1 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [-1,-3,-2,-4,-6,-5] => ?
 => ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [-1,-3,-2,-5,-4,-6] => ?
 => ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 2 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [-1,-3,-2,-6,-5,-4] => ?
 => ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [-1,-3,-4,-2,-5,-6] => ?
 => ? = 1 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 2 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [-1,-3,-4,-5,-2,-6] => ?
 => ? = 1 - 1
[1,3,4,6,5,2] => [1,3,4,6,5,2] => [-1,-3,-4,-6,-5,-2] => ?
 => ? = 1 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [-1,-3,-5,-2,-4,-6] => ?
 => ? = 1 - 1
[1,3,5,4,2,6] => [1,3,5,4,2,6] => [-1,-3,-5,-4,-2,-6] => ?
 => ? = 1 - 1
[1,3,5,4,6,2] => [1,3,5,4,6,2] => [-1,-3,-5,-4,-6,-2] => ?
 => ? = 1 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ? => ?
 => ? = 1 - 1
[1,3,6,4,2,5] => [1,3,6,4,2,5] => ? => ?
 => ? = 1 - 1
[1,3,6,4,5,2] => [1,3,6,4,5,2] => [-1,-3,-6,-4,-5,-2] => ?
 => ? = 1 - 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [-1,-3,-6,-5,-4,-2] => ?
 => ? = 2 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [-1,-4,-2,-3,-5,-6] => ?
 => ? = 1 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 2 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [-1,-4,-2,-5,-3,-6] => ?
 => ? = 1 - 1
[1,4,2,6,5,3] => [1,4,2,6,5,3] => ? => ?
 => ? = 1 - 1
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [-1,-4,-3,-2,-5,-6] => ?
 => ? = 1 - 1

Description

This is the number of standard Young tableaux of the given shifted shape. For an integer partition

$\lambda = (\lambda_1,\dots,\lambda_k)$ , the shifted diagram is obtained by moving the

$i$ -th row in the diagram

$i-1$ boxes to the right, i.e.,

$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$ In particular, this statistic is zero if and only if

$\lambda_{i+1} = \lambda_i$ for some

$i$ .

Matching statistic: St001625
(load all 2 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 13%

Values

[1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 1
[1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 2
[1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,3,5,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1
[1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,4,3,2,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 1
[1,4,3,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1
[1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2
[1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1
[1,5,3,2,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1
[1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
 => ? = 1
[1,5,4,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
 => ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[2,1,4,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2
[2,1,5,3,4] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2
[2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 2
[2,3,1,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,1,5,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,3,1,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,4,5,1,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,5,3,4,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,5,4,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,1,2,4,5] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2
[3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[3,2,4,1,5] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,5,4,1] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,4,1,2,5] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[3,4,1,5,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2
[3,4,5,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,5,1,2,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2
[3,5,1,4,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2
[3,5,4,1,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2
[4,1,3,2,5] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[4,1,5,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
 => ? = 2
[4,2,1,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[4,2,3,5,1] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,2,5,1,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[4,3,2,5,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[4,3,5,1,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2
[4,5,1,3,2] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 2
[4,5,2,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2
[4,5,3,1,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2
[2,3,4,1,5,6] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,4,1,6,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,5,4,1,6] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,6,4,5,1] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,6,5,4,1] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,3,5,1,6] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,4,3,6,5,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,5,3,4,6,1] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,5,4,3,6,1] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,4,5,1,6] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,4,6,5,1] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,5,4,6,1] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,2,3,5,6,1] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,2,5,6,1] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,3,5,6,1,7] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,4,3,6,5,7,1] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,4,5,6,1,7] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,5,4,6,7,1] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,2,3,5,6,7,1] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,2,5,6,7,1] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2

Description

The Möbius invariant of a lattice. The '''Möbius invariant''' of a lattice

$L$ is the value of the Möbius function applied to least and greatest element, that is

$\mu(L)=\mu_L(\hat{0},\hat{1})$ , where

$\hat{0}$ is the least element of

$L$ and

$\hat{1}$ is the greatest element of

$L$ . For the definition of the Möbius function, see [[St000914]].

Matching statistic: St001621
(load all 2 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 13%

Values

[1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1 + 1
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 + 1
[1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1 + 1
[1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1 + 1
[1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1 + 1
[1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1 + 1
[1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 1 + 1
[1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1 + 1
[1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 2 + 1
[1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1 + 1
[1,3,5,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1 + 1
[1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1 + 1
[1,4,3,2,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 1 + 1
[1,4,3,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1 + 1
[1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2 + 1
[1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1 + 1
[1,5,3,2,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 1 + 1
[1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
 => ? = 1 + 1
[1,5,4,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
 => ? = 2 + 1
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 1
[2,1,5,3,4] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 1
[2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 2 + 1
[2,3,1,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,1,5,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,4,3,1,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,5,1,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,5,3,4,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,5,4,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,1,2,4,5] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 + 1
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2 + 1
[3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[3,2,4,1,5] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,5,4,1] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,1,2,5] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 + 1
[3,4,1,5,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2 + 1
[3,4,5,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,5,1,2,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 + 1
[3,5,1,4,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2 + 1
[3,5,4,1,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2 + 1
[4,1,3,2,5] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1 + 1
[4,1,5,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
 => ? = 2 + 1
[4,2,1,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 + 1
[4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[4,2,3,5,1] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,5,1,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 + 1
[4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,3,2,5,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,3,5,1,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 2 + 1
[4,5,1,3,2] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 2 + 1
[4,5,2,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 + 1
[4,5,3,1,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2 + 1
[2,3,4,1,5,6] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1,6,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,5,4,1,6] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,6,4,5,1] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,6,5,4,1] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,4,3,5,1,6] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,6,5,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,5,3,4,6,1] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,5,4,3,6,1] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,4,5,1,6] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,6,5,1] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,5,4,6,1] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,5,6,1] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,5,6,1] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,4,3,5,6,1,7] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,6,5,7,1] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,4,5,6,1,7] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,5,4,6,7,1] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,5,6,7,1] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,5,6,7,1] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001964The interval resolution global dimension of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001866The nesting alignments of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.