- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000356

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0]
 => [1] => [1] => 0
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[1,1]
 => [1,1,0,0]
 => [2,1] => [2,1] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,3,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,3,4,2] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,4,5,3] => 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,4,1,2] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,3,5,2,4] => 2
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,3,4,1] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => 2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,5,2,3] => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [1,2,4,6,3,5] => 2
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,5,1,2,4] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [1,3,6,2,4,5] => 3
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => 4
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,3,5,1,4] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [2,6,1,3,4,5] => 3
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => 5
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 0
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,6,7] => 2
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 2
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => 2

Description

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$13\!\!-\!\!2$ .

Matching statistic: St001683

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 40%

Values

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

Description

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.

Matching statistic: St001685

Mapping

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 40%

Values

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

Description

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

Matching statistic: St000355

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$21\!\!-\!\!3$ .

Matching statistic: St001744

Mapping

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let

$\nu$ be a (partial) permutation of

$[k]$ with

$m$ letters together with dashes between some of its letters. An occurrence of

$\nu$ in a permutation

$\tau$ is a subsequence

$\tau_{a_1},\dots,\tau_{a_m}$ such that

$a_i + 1 = a_{i+1}$ whenever there is a dash between the

$i$ -th and the

$(i+1)$ -st letter of

$\nu$ , which is order isomorphic to

$\nu$ . Thus,

$\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size

$k$ consists of such a generalized vincular pattern

$\nu$ and arrows

$b_1\to c_1, b_2\to c_2,\dots$ , such that precisely the numbers

$1,\dots,k$ appear in the vincular pattern and the arrows. Let

$\Phi$ be the map [[Mp00087]]. Let

$\tau$ be a permutation and

$\sigma = \Phi(\tau)$ . Then a subsequence

$w = (x_{a_1},\dots,x_{a_m})$ of

$\tau$ is an occurrence of the arrow pattern if

$w$ is an occurrence of

$\nu$ , for each arrow

$b\to c$ we have

$\sigma(x_b) = x_c$ and

$x_1 < x_2 < \dots < x_k$ .

Matching statistic: St001435

Mapping

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%

Values

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

Description

The number of missing boxes in the first row.

Matching statistic: St001575

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001575: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 30%

Values

[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 0
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 0
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 0
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 0
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => ? = 0
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 0
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? = 0
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? = 0
[7,1]
 => 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[6,2]
 => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[6,1,1]
 => 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,2,1]
 => 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ? = 0
[4,3,1]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2]
 => 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1]
 => 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,1,1,1,1,1,1]
 => 101111110 => [1,2,1,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6
[1,1,1,1,1,1,1,1]
 => 111111110 => [1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[9]
 => 1000000000 => [1,10] => ([(9,10)],11)
 => ? = 0
[]
 =>  => [1] => ([],1)
 => 0

Description

The minimal number of edges to add or remove to make a graph edge transitive. A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.

Matching statistic: St000455

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 10%

Values

[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 0
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 0
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 0
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 0
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 0
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 0
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? = 0
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? = 0
[7,1]
 => 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[6,2]
 => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[6,1,1]
 => 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,2,1]
 => 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 0
[4,3,1]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2]
 => 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1]
 => 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,1,1,1,1,1,1]
 => 101111110 => [1,2,1,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6
[1,1,1,1,1,1,1,1]
 => 111111110 => [1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0

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.