- FindStat

Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001673
(load all 2 compositions to match this statistic)

Mapping

St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.

Matching statistic: St000486
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycles of length at least 3 of a permutation.

Matching statistic: St001359
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is

$2^k$ where

$k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes,

$f(n)$ , is

$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$

Matching statistic: St000455

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0 - 1
[2] => [[2],[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0 - 1
[1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0 - 1
[1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0 - 1
[1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0 - 1
[1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 - 1
[1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 - 1
[3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0 - 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 1 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,4] => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 1 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,4,1] => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 1 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 - 1
[4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 1 - 1
[4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 - 1
[5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0 - 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ? = 0 - 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 1 - 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 1 - 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => ? = 1 - 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 0 - 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 1 - 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 0 - 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => ? = 1 - 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: St001624

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%

Values

[1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2] => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 1 + 1
[1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 1 + 1
[1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
 => ? = 1 + 1
[1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 0 + 1
[1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 0 + 1
[2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
 => ? = 1 + 1
[2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 1 + 1
[3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 + 1
[3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 1 + 1
[4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 1 + 1
[5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 1 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(0,6),(1,4),(1,13),(2,11),(3,9),(4,3),(4,12),(5,10),(6,1),(6,10),(8,7),(9,7),(10,2),(10,13),(11,8),(12,8),(12,9),(13,11),(13,12)],14)
 => ? = 1 + 1
[1,1,1,3] => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 1 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(1,10),(2,11),(3,4),(3,14),(4,8),(5,2),(5,13),(6,3),(6,13),(8,9),(9,7),(10,7),(11,1),(11,12),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 0 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,4),(0,5),(1,14),(2,3),(2,15),(3,12),(4,1),(4,13),(5,2),(5,13),(6,9),(6,11),(7,17),(8,17),(9,16),(10,8),(10,16),(11,7),(11,16),(12,7),(12,8),(13,6),(13,14),(13,15),(14,9),(14,10),(15,10),(15,11),(15,12),(16,17)],18)
 => ? = 2 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,13),(2,14),(3,5),(3,6),(3,14),(4,7),(4,9),(5,10),(5,12),(6,4),(6,11),(6,12),(7,16),(9,16),(10,1),(10,15),(11,9),(11,15),(12,7),(12,15),(13,8),(14,10),(14,11),(15,13),(15,16),(16,8)],17)
 => ? = 1 + 1
[1,1,4] => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
 => ? = 1 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(0,3),(1,14),(2,1),(2,15),(3,5),(3,6),(3,15),(4,7),(4,8),(5,9),(5,11),(6,9),(6,12),(7,17),(8,17),(9,18),(10,16),(11,10),(11,18),(12,4),(12,13),(12,18),(13,7),(13,16),(14,10),(14,13),(15,11),(15,12),(15,14),(16,17),(18,8),(18,16)],19)
 => ? = 2 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ?
 => ? = 0 + 1
[1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,4),(0,5),(1,14),(2,3),(2,15),(3,12),(4,1),(4,13),(5,2),(5,13),(6,9),(6,11),(7,17),(8,17),(9,16),(10,8),(10,16),(11,7),(11,16),(12,7),(12,8),(13,6),(13,14),(13,15),(14,9),(14,10),(15,10),(15,11),(15,12),(16,17)],18)
 => ? = 1 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,10),(2,13),(3,5),(3,6),(3,14),(4,2),(4,14),(5,8),(5,11),(6,8),(6,12),(7,16),(8,15),(9,1),(9,16),(11,7),(11,15),(12,9),(12,15),(13,7),(13,9),(14,11),(14,12),(14,13),(15,16),(16,10)],17)
 => ? = 1 + 1
[1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(0,3),(1,14),(2,1),(2,15),(3,5),(3,6),(3,15),(4,7),(4,8),(5,9),(5,11),(6,9),(6,12),(7,17),(8,17),(9,18),(10,16),(11,10),(11,18),(12,4),(12,13),(12,18),(13,7),(13,16),(14,10),(14,13),(15,11),(15,12),(15,14),(16,17),(18,8),(18,16)],19)
 => ? = 1 + 1
[1,4,1] => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ([(0,2),(0,3),(1,8),(2,13),(3,5),(3,6),(3,13),(4,7),(4,9),(5,10),(5,12),(6,4),(6,11),(6,12),(7,15),(9,1),(9,15),(10,14),(11,9),(11,14),(12,7),(12,14),(13,10),(13,11),(14,15),(15,8)],16)
 => ? = 0 + 1
[1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 1 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 1 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ([(0,2),(0,3),(1,8),(2,13),(3,5),(3,6),(3,13),(4,7),(4,9),(5,10),(5,12),(6,4),(6,11),(6,12),(7,15),(9,1),(9,15),(10,14),(11,9),(11,14),(12,7),(12,14),(13,10),(13,11),(14,15),(15,8)],16)
 => ? = 0 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(1,13),(2,8),(2,10),(3,7),(3,12),(4,11),(4,15),(5,11),(5,14),(6,3),(6,14),(6,15),(7,1),(7,16),(8,18),(10,18),(11,17),(12,8),(12,16),(13,9),(14,7),(14,17),(15,2),(15,12),(15,17),(16,13),(16,18),(17,10),(17,16),(18,9)],19)
 => ? = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,13),(2,14),(3,5),(3,6),(3,14),(4,7),(4,9),(5,10),(5,12),(6,4),(6,11),(6,12),(7,16),(9,16),(10,1),(10,15),(11,9),(11,15),(12,7),(12,15),(13,8),(14,10),(14,11),(15,13),(15,16),(16,8)],17)
 => ? = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(1,12),(2,13),(3,8),(3,9),(4,10),(4,15),(5,10),(5,14),(6,3),(6,14),(6,15),(8,17),(9,1),(9,17),(10,2),(10,16),(11,7),(12,7),(13,11),(14,9),(14,16),(15,8),(15,16),(16,13),(16,17),(17,11),(17,12)],18)
 => ? = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ?
 => ? = 0 + 1
[2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(1,13),(2,8),(2,10),(3,7),(3,12),(4,11),(4,15),(5,11),(5,14),(6,3),(6,14),(6,15),(7,1),(7,16),(8,18),(10,18),(11,17),(12,8),(12,16),(13,9),(14,7),(14,17),(15,2),(15,12),(15,17),(16,13),(16,18),(17,10),(17,16),(18,9)],19)
 => ? = 1 + 1
[2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(0,6),(1,4),(1,13),(2,11),(3,9),(4,3),(4,12),(5,10),(6,1),(6,10),(8,7),(9,7),(10,2),(10,13),(11,8),(12,8),(12,9),(13,11),(13,12)],14)
 => ? = 1 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 1 + 1
[3,1,2] => [[4,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,10),(2,13),(3,5),(3,6),(3,14),(4,2),(4,14),(5,8),(5,11),(6,8),(6,12),(7,16),(8,15),(9,1),(9,16),(11,7),(11,15),(12,9),(12,15),(13,7),(13,9),(14,11),(14,12),(14,13),(15,16),(16,10)],17)
 => ? = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(1,12),(2,13),(3,8),(3,9),(4,10),(4,15),(5,10),(5,14),(6,3),(6,14),(6,15),(8,17),(9,1),(9,17),(10,2),(10,16),(11,7),(12,7),(13,11),(14,9),(14,16),(15,8),(15,16),(16,13),(16,17),(17,11),(17,12)],18)
 => ? = 1 + 1
[3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(1,10),(2,11),(3,4),(3,14),(4,8),(5,2),(5,13),(6,3),(6,13),(8,9),(9,7),(10,7),(11,1),(11,12),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 1 + 1
[4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
 => ? = 1 + 1
[5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 1 + 1
[6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,6),(0,7),(1,12),(2,13),(3,5),(3,18),(4,9),(5,4),(5,14),(6,2),(6,16),(7,3),(7,16),(9,10),(10,8),(11,8),(12,11),(13,1),(13,15),(14,9),(14,17),(15,12),(15,17),(16,13),(16,18),(17,10),(17,11),(18,14),(18,15)],19)
 => ? = 1 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,6),(0,7),(1,13),(2,4),(2,17),(3,5),(3,18),(4,9),(5,12),(6,2),(6,15),(7,3),(7,15),(9,10),(10,11),(11,8),(12,1),(12,16),(13,8),(14,10),(14,16),(15,17),(15,18),(16,11),(16,13),(17,9),(17,14),(18,12),(18,14)],19)
 => ? = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
 => ?
 => ? = 1 + 1

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .

Matching statistic: St000630

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%

Values

[1,1] => [1,0,1,0]
 => 1010 => 1111 => 1 = 0 + 1
[2] => [1,1,0,0]
 => 1100 => 1001 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 101010 => 111111 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => 101100 => 111001 => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
 => 110010 => 100111 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => 111000 => 101101 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 11111111 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 10101100 => 11111001 => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 10110010 => 11100111 => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 10111000 => 11101101 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 11001010 => 10011111 => 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 11100010 => 10110111 => 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 11110000 => 10100101 => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1111111111 => ? = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1111111001 => ? = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1111100111 => ? = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1111101101 => ? = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1110011111 => ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1110011001 => ? = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1110110111 => ? = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1110100101 => ? = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001111111 => ? = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001111001 => ? = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100111 => ? = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001101101 => ? = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1011011111 => ? = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1011011001 => ? = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1010010111 => ? = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 111111111111 => ? = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 111111111001 => ? = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 111111100111 => ? = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 111111101101 => ? = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 111110011111 => ? = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 111110011001 => ? = 2 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => 111110110111 => ? = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => 111110100101 => ? = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 101100101010 => 111001111111 => ? = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 101100101100 => 111001111001 => ? = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 101100110010 => 111001100111 => ? = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => 111001101101 => ? = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 101110001010 => 111011011111 => ? = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 101110001100 => 111011011001 => ? = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 101111000010 => 111010010111 => ? = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 111010110101 => ? = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 110010101010 => 100111111111 => ? = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 110010101100 => 100111111001 => ? = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 110010110010 => 100111100111 => ? = 2 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 110010111000 => 100111101101 => ? = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 110011001010 => 100110011111 => ? = 2 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 110011001100 => 100110011001 => ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 110011100010 => 100110110111 => ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 100110100101 => ? = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 111000101010 => 101101111111 => ? = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 111000101100 => 101101111001 => ? = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 111000110010 => 101101100111 => ? = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 101101101101 => ? = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 111100001010 => 101001011111 => ? = 1 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 111100001100 => 101001011001 => ? = 1 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => 101011010111 => ? = 1 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 111111000000 => 101010010101 => ? = 0 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 10101010101010 => 11111111111111 => ? = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 10101011001010 => 11111110011111 => ? = 1 + 1

Description

The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word

$w=a_1,\dots,a_n$ is any list of factors

$p_1,\dots,p_k$ such that

$w=p_1\dots p_k$ and each

$p_i$ is a palindrome, i.e. coincides with itself read backwards.

Matching statistic: St001569

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%

Values

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

Description

The maximal modular displacement of a permutation. This is

$\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation

$\pi$ of

$\{1,\dots,n\}$ .