- FindStat

Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001137

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

Matching statistic: St001141
(load all 17 compositions to match this statistic)

Mapping

St001141: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => ? = 0
[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 0

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

Matching statistic: St001060
(load all 2 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%

Values

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

Description

The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.

Matching statistic: St000648

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of 2-excedences of a permutation. This is the number of positions

$1\leq i\leq n$ such that

$\sigma(i)=i+2$ .

Matching statistic: St001878
(load all 4 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%

Values

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

Description

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

Matching statistic: St001663
(load all 4 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001663: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the Hertzsprung pattern 132 in a permutation. A Hertzsprung occurrence of the pattern

$\tau=(\tau_1,\dots,\tau_k)$ in a permutation

$\pi$ is a factor

$\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of

$\pi$ such that

$\pi_{i+j-1} - \tau_j$ is constant for

$1\leq j\leq k$ .

Matching statistic: St001876
(load all 2 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of 2-regular simple modules in the incidence algebra of the lattice.

Matching statistic: St001877
(load all 2 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

Number of indecomposable injective modules with projective dimension 2.

Matching statistic: St001545

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The second Elser number of a connected graph. For a connected graph

$G$ the

$k$ -th Elser number is

$els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k$ where the sum is over all nuclei of

$G$ , that is, the connected subgraphs of

$G$ whose vertex set is a vertex cover of

$G$ . It is clear that this number is even. It was shown in [1] that it is non-negative.

Matching statistic: St001330
(load all 2 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000264The girth of a graph, which is not a tree. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000488The number of cycles of a permutation of length at most 2.