- FindStat

Your data matches 88 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000445

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000455
(load all 3 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 28%●distinct values known / distinct values provided: 17%

Values

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

Description

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

Matching statistic: St001677
(load all 2 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001677: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 0
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 0
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 0
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 0
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 0
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 0
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 0
[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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 0
[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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
[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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 0
[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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
[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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0
[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)
 => ?
 => ? = 2
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0

Description

The number of non-degenerate subsets of a lattice whose meet is the bottom element. A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.

Matching statistic: St001845
(load all 9 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 0
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 0
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 0
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 0
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 0
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 0
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 0
[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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 0
[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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 0
[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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
[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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 0
[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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
[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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0
[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)
 => ?
 => ? = 2
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0
[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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0

Description

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is $0$.

Matching statistic: St001613
(load all 3 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 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)
 => ?
 => ? = 2 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 1

Description

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.

Matching statistic: St001681
(load all 2 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001681: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 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)
 => ?
 => ? = 2 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 1

Description

The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.

Matching statistic: St001719
(load all 51 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 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)
 => ?
 => ? = 2 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 1

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.

Matching statistic: St001881
(load all 3 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 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)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 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)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 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)
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => 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)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => ? = 1 + 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)
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => ? = 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)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 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)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
 => ? = 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)
 => ?
 => ? = 2 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 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)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
 => ? = 1 + 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 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)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 0 + 1

Description

The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

Matching statistic: St000069

Mapping

Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000069: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 1 = 0 + 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,4],[2,3,4],[3,4],[4]]
 => ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ([],1)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,4,5],[4,5],[5]]
 => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
 => ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
 => ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,2],[2,2,2,3],[3,4,5],[4,5],[5]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
 => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
 => ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [[1,1,1,1,2],[2,3,4,5],[3,4,5],[4,5],[5]]
 => ([(0,5),(0,14),(0,15),(1,26),(2,9),(2,25),(3,13),(3,24),(4,16),(5,11),(5,12),(6,69),(7,64),(8,49),(9,6),(9,52),(10,20),(10,59),(11,19),(11,65),(12,17),(12,65),(13,18),(13,22),(13,71),(14,2),(14,29),(15,3),(15,29),(16,39),(16,55),(17,56),(17,68),(18,57),(18,66),(19,58),(19,67),(20,51),(20,53),(21,40),(21,41),(22,57),(22,72),(23,28),(23,54),(23,70),(24,58),(24,71),(25,52),(25,56),(26,27),(26,69),(26,72),(27,45),(27,60),(28,34),(28,48),(28,50),(29,1),(30,76),(31,75),(32,77),(33,74),(33,80),(34,73),(34,78),(35,73),(35,77),(36,82),(37,79),(38,80),(39,74),(40,7),(40,76),(41,8),(41,76),(42,55),(42,79),(43,59),(44,39),(45,53),(45,83),(46,33),(46,81),(47,32),(47,83),(48,42),(48,78),(49,44),(50,46),(50,73),(51,44),(52,10),(52,43),(53,62),(54,34),(54,75),(55,36),(55,74),(56,43),(57,63),(58,61),(59,49),(59,51),(60,35),(60,50),(60,83),(61,31),(61,70),(62,33),(62,38),(63,32),(63,35),(64,37),(64,42),(65,21),(65,67),(65,68),(66,31),(66,54),(67,30),(67,40),(68,30),(68,41),(69,45),(69,47),(70,48),(70,64),(70,75),(71,23),(71,61),(71,66),(72,47),(72,60),(72,63),(73,81),(74,82),(75,37),(75,78),(76,4),(77,38),(77,81),(78,79),(79,36),(80,82),(81,80),(83,46),(83,62),(83,77)],84)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,4,5],[4,5],[5]]
 => ([(0,7),(0,8),(0,11),(1,25),(2,10),(2,24),(3,5),(3,12),(3,26),(4,28),(5,6),(5,29),(6,4),(6,27),(7,2),(7,19),(8,3),(8,18),(9,15),(9,16),(10,13),(10,23),(11,18),(11,19),(12,22),(12,23),(12,29),(13,31),(14,30),(15,30),(16,30),(17,20),(18,26),(19,24),(20,14),(21,16),(22,25),(22,31),(23,17),(23,31),(24,13),(25,9),(25,21),(26,1),(26,22),(27,20),(27,28),(28,14),(28,15),(29,17),(29,27),(31,21)],32)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
 => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,4,5],[4,5],[5]]
 => ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
 => ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
 => ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
 => ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [[1,1,2,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
 => 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 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
 => 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 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,1],[2,2,2,2,3],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,5),(0,14),(0,15),(1,26),(2,9),(2,25),(3,13),(3,24),(4,16),(5,11),(5,12),(6,69),(7,64),(8,49),(9,6),(9,52),(10,20),(10,59),(11,19),(11,65),(12,17),(12,65),(13,18),(13,22),(13,71),(14,2),(14,29),(15,3),(15,29),(16,39),(16,55),(17,56),(17,68),(18,57),(18,66),(19,58),(19,67),(20,51),(20,53),(21,40),(21,41),(22,57),(22,72),(23,28),(23,54),(23,70),(24,58),(24,71),(25,52),(25,56),(26,27),(26,69),(26,72),(27,45),(27,60),(28,34),(28,48),(28,50),(29,1),(30,76),(31,75),(32,77),(33,74),(33,80),(34,73),(34,78),(35,73),(35,77),(36,82),(37,79),(38,80),(39,74),(40,7),(40,76),(41,8),(41,76),(42,55),(42,79),(43,59),(44,39),(45,53),(45,83),(46,33),(46,81),(47,32),(47,83),(48,42),(48,78),(49,44),(50,46),(50,73),(51,44),(52,10),(52,43),(53,62),(54,34),(54,75),(55,36),(55,74),(56,43),(57,63),(58,61),(59,49),(59,51),(60,35),(60,50),(60,83),(61,31),(61,70),(62,33),(62,38),(63,32),(63,35),(64,37),(64,42),(65,21),(65,67),(65,68),(66,31),(66,54),(67,30),(67,40),(68,30),(68,41),(69,45),(69,47),(70,48),(70,64),(70,75),(71,23),(71,61),(71,66),(72,47),(72,60),(72,63),(73,81),(74,82),(75,37),(75,78),(76,4),(77,38),(77,81),(78,79),(79,36),(80,82),(81,80),(83,46),(83,62),(83,77)],84)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]]
 => ([(0,7),(0,8),(0,11),(1,25),(2,10),(2,24),(3,5),(3,12),(3,26),(4,28),(5,6),(5,29),(6,4),(6,27),(7,2),(7,19),(8,3),(8,18),(9,15),(9,16),(10,13),(10,23),(11,18),(11,19),(12,22),(12,23),(12,29),(13,31),(14,30),(15,30),(16,30),(17,20),(18,26),(19,24),(20,14),(21,16),(22,25),(22,31),(23,17),(23,31),(24,13),(25,9),(25,21),(26,1),(26,22),(27,20),(27,28),(28,14),(28,15),(29,17),(29,27),(31,21)],32)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
 => ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
 => ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,1],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]]
 => 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 => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [[1,1,1,1,1,1],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]]
 => 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 => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
 => ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
 => ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
 => ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
 => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
 => ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,5,6],[5,6],[6]]
 => ([(0,7),(0,8),(1,13),(2,11),(3,10),(3,11),(4,5),(4,10),(5,1),(5,12),(6,2),(7,6),(8,9),(9,3),(9,4),(10,12),(10,16),(11,16),(12,13),(12,15),(13,14),(15,14),(16,15)],17)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,4),(0,8),(1,13),(2,10),(3,10),(3,11),(4,12),(5,9),(6,1),(6,11),(7,2),(8,5),(8,12),(9,3),(9,6),(10,14),(11,13),(11,14),(12,7),(13,15),(14,15)],16)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
 => ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
 => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,2,3],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,5),(0,14),(0,15),(1,25),(2,9),(2,26),(3,13),(3,24),(4,16),(5,10),(5,11),(6,79),(7,73),(8,55),(9,6),(9,62),(10,17),(10,74),(11,19),(11,74),(12,20),(12,67),(13,18),(13,22),(13,80),(14,2),(14,31),(15,3),(15,31),(16,40),(16,64),(17,66),(17,77),(18,81),(18,84),(19,65),(19,78),(20,61),(20,63),(21,41),(21,42),(22,27),(22,81),(22,83),(23,29),(23,75),(23,76),(24,66),(24,80),(25,28),(25,79),(25,84),(26,62),(26,65),(27,48),(27,70),(28,50),(28,69),(29,58),(29,59),(29,60),(30,93),(30,96),(31,1),(32,89),(33,88),(34,87),(34,94),(35,90),(36,90),(36,93),(37,92),(38,98),(39,97),(40,87),(41,7),(41,89),(42,8),(42,89),(43,40),(44,64),(44,103),(45,39),(45,94),(46,37),(46,103),(47,38),(47,95),(48,72),(49,53),(49,100),(50,61),(50,100),(51,67),(52,34),(52,91),(53,35),(53,101),(54,33),(54,102),(55,43),(56,30),(56,88),(57,36),(57,86),(57,101),(58,44),(58,85),(59,52),(59,86),(60,30),(60,85),(60,86),(61,71),(62,12),(62,51),(63,43),(64,47),(64,87),(65,51),(66,68),(67,55),(67,63),(68,54),(68,76),(69,57),(69,59),(69,100),(70,33),(70,56),(71,34),(71,45),(72,35),(72,36),(73,44),(73,46),(74,21),(74,77),(74,78),(75,56),(75,60),(75,102),(76,58),(76,73),(76,102),(77,32),(77,41),(78,32),(78,42),(79,49),(79,50),(80,23),(80,68),(80,83),(81,48),(81,82),(82,53),(82,57),(82,72),(83,54),(83,70),(83,75),(84,49),(84,69),(84,82),(85,96),(85,103),(86,91),(86,93),(87,95),(88,37),(88,96),(89,4),(90,39),(90,99),(91,94),(91,99),(92,38),(93,99),(94,95),(94,97),(95,98),(96,92),(97,98),(99,97),(100,52),(100,71),(100,101),(101,45),(101,90),(101,91),(102,46),(102,85),(102,88),(103,47),(103,92)],104)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
 => ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,6],[6]]
 => ([(0,4),(0,8),(1,10),(2,11),(3,10),(3,11),(4,12),(5,6),(5,13),(6,9),(7,1),(8,5),(8,12),(9,2),(9,3),(10,14),(11,14),(12,13),(13,7)],15)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]]
 => ([(0,6),(0,7),(0,12),(1,29),(2,11),(2,28),(3,4),(3,13),(3,30),(4,5),(4,32),(5,8),(5,9),(5,34),(6,2),(6,21),(7,3),(7,20),(8,18),(8,27),(9,18),(9,33),(10,17),(10,26),(11,14),(11,25),(12,20),(12,21),(13,24),(13,25),(13,32),(14,39),(15,37),(16,36),(16,37),(17,36),(18,35),(19,38),(20,30),(21,28),(22,17),(23,31),(24,29),(24,39),(25,23),(25,39),(26,19),(26,36),(27,15),(27,35),(28,14),(29,10),(29,22),(30,1),(30,24),(31,15),(31,16),(32,23),(32,34),(33,16),(33,26),(33,35),(34,27),(34,31),(34,33),(35,19),(35,37),(36,38),(37,38),(39,22)],40)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
 => ? = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,5,6],[5,6],[6]]
 => ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,6),(3,7),(4,36),(5,35),(6,17),(6,43),(7,10),(7,43),(8,9),(8,42),(9,11),(9,44),(10,40),(11,41),(12,28),(12,30),(13,21),(13,34),(14,20),(14,25),(15,31),(15,32),(16,29),(16,42),(17,29),(17,39),(18,45),(19,49),(20,12),(20,49),(21,47),(23,46),(24,5),(25,4),(25,49),(26,21),(26,45),(27,18),(27,48),(28,38),(29,37),(30,24),(31,28),(31,48),(32,13),(32,26),(32,48),(33,35),(33,46),(34,33),(34,47),(35,22),(36,24),(37,27),(37,31),(38,23),(38,33),(39,19),(39,20),(40,19),(40,25),(41,18),(41,26),(42,15),(42,37),(42,44),(43,14),(43,39),(43,40),(44,27),(44,32),(44,41),(45,23),(45,47),(46,22),(47,46),(48,34),(48,38),(48,45),(49,30),(49,36)],50)
 => ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
 => ([(0,5),(0,14),(0,15),(1,26),(2,9),(2,25),(3,13),(3,24),(4,16),(5,11),(5,12),(6,69),(7,64),(8,49),(9,6),(9,52),(10,20),(10,59),(11,19),(11,65),(12,17),(12,65),(13,18),(13,22),(13,71),(14,2),(14,29),(15,3),(15,29),(16,39),(16,55),(17,56),(17,68),(18,57),(18,66),(19,58),(19,67),(20,51),(20,53),(21,40),(21,41),(22,57),(22,72),(23,28),(23,54),(23,70),(24,58),(24,71),(25,52),(25,56),(26,27),(26,69),(26,72),(27,45),(27,60),(28,34),(28,48),(28,50),(29,1),(30,76),(31,75),(32,77),(33,74),(33,80),(34,73),(34,78),(35,73),(35,77),(36,82),(37,79),(38,80),(39,74),(40,7),(40,76),(41,8),(41,76),(42,55),(42,79),(43,59),(44,39),(45,53),(45,83),(46,33),(46,81),(47,32),(47,83),(48,42),(48,78),(49,44),(50,46),(50,73),(51,44),(52,10),(52,43),(53,62),(54,34),(54,75),(55,36),(55,74),(56,43),(57,63),(58,61),(59,49),(59,51),(60,35),(60,50),(60,83),(61,31),(61,70),(62,33),(62,38),(63,32),(63,35),(64,37),(64,42),(65,21),(65,67),(65,68),(66,31),(66,54),(67,30),(67,40),(68,30),(68,41),(69,45),(69,47),(70,48),(70,64),(70,75),(71,23),(71,61),(71,66),(72,47),(72,60),(72,63),(73,81),(74,82),(75,37),(75,78),(76,4),(77,38),(77,81),(78,79),(79,36),(80,82),(81,80),(83,46),(83,62),(83,77)],84)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]]
 => ([(0,14),(0,15),(0,28),(1,26),(2,13),(2,25),(3,16),(4,7),(4,53),(5,54),(6,72),(7,74),(8,2),(8,30),(9,21),(9,52),(10,17),(10,73),(11,19),(11,64),(12,20),(12,65),(13,18),(13,23),(13,78),(14,12),(14,47),(15,9),(15,66),(16,40),(16,60),(17,63),(17,76),(18,62),(18,75),(19,56),(19,58),(20,43),(20,59),(21,43),(21,57),(22,41),(22,42),(23,62),(23,79),(24,29),(24,61),(24,77),(25,63),(25,78),(26,27),(26,74),(26,79),(27,46),(27,67),(28,8),(28,47),(28,66),(29,35),(29,51),(29,55),(30,1),(31,84),(32,82),(33,81),(33,87),(34,83),(35,80),(35,85),(36,80),(36,82),(37,87),(38,86),(39,89),(40,81),(41,6),(41,84),(42,5),(42,84),(43,90),(44,40),(45,60),(45,86),(46,56),(46,91),(47,10),(47,65),(48,64),(49,33),(49,88),(50,32),(50,91),(51,45),(51,85),(52,4),(52,57),(53,11),(53,48),(54,44),(55,49),(55,80),(56,69),(57,53),(57,90),(58,44),(59,70),(59,90),(60,39),(60,81),(61,35),(61,83),(62,71),(63,68),(64,54),(64,58),(65,59),(65,73),(66,30),(66,52),(67,36),(67,55),(67,91),(68,34),(68,77),(69,33),(69,37),(70,31),(70,42),(71,32),(71,36),(72,38),(72,45),(73,22),(73,70),(73,76),(74,46),(74,50),(75,34),(75,61),(76,31),(76,41),(77,51),(77,72),(77,83),(78,24),(78,68),(78,75),(79,50),(79,67),(79,71),(80,88),(81,89),(82,37),(82,88),(83,38),(83,85),(84,3),(85,86),(86,39),(87,89),(88,87),(90,48),(91,49),(91,69),(91,82)],92)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
 => ([(0,13),(0,14),(0,30),(1,28),(2,95),(3,94),(4,77),(5,10),(5,110),(6,70),(7,8),(7,31),(7,71),(8,12),(8,99),(9,23),(9,100),(10,22),(10,111),(11,20),(11,98),(12,17),(12,18),(12,112),(13,7),(13,64),(14,5),(14,27),(14,63),(15,53),(15,87),(16,82),(16,104),(17,55),(17,107),(18,55),(18,105),(19,52),(19,80),(20,78),(20,84),(21,79),(21,86),(22,54),(22,108),(23,54),(23,106),(24,51),(24,83),(25,29),(25,102),(25,103),(26,33),(26,85),(26,109),(27,32),(27,101),(27,110),(28,50),(28,81),(29,62),(29,88),(30,63),(30,64),(31,65),(31,68),(31,99),(32,68),(32,69),(32,93),(33,36),(33,42),(33,76),(34,118),(34,134),(35,121),(35,142),(36,118),(36,141),(37,3),(37,117),(38,116),(38,126),(39,115),(39,126),(40,121),(40,129),(41,114),(41,125),(42,118),(42,127),(43,117),(44,133),(45,132),(46,123),(47,128),(48,124),(49,125),(50,114),(51,6),(51,139),(52,2),(52,115),(53,16),(53,138),(54,119),(55,136),(56,84),(56,140),(57,60),(58,73),(58,135),(59,47),(59,140),(60,50),(61,51),(61,116),(62,56),(62,113),(63,9),(63,101),(64,71),(65,77),(65,137),(66,102),(67,38),(67,138),(68,89),(68,137),(69,90),(69,137),(70,57),(71,4),(71,65),(72,85),(72,129),(73,81),(73,123),(74,34),(74,122),(75,41),(75,130),(76,58),(76,127),(76,141),(77,25),(77,66),(78,60),(79,82),(79,120),(80,45),(80,115),(81,44),(81,114),(82,97),(83,37),(83,139),(84,91),(85,42),(85,131),(86,26),(86,72),(86,120),(87,24),(87,61),(87,138),(88,34),(88,36),(88,113),(89,79),(89,92),(90,39),(90,52),(91,41),(91,49),(92,35),(92,40),(92,120),(93,53),(93,67),(94,46),(94,73),(95,56),(95,59),(96,58),(96,94),(96,124),(97,48),(97,96),(98,57),(98,78),(99,21),(99,89),(99,112),(100,19),(100,90),(100,106),(101,69),(101,100),(102,62),(102,95),(102,142),(103,74),(103,88),(103,142),(104,37),(104,43),(105,35),(105,103),(105,136),(106,39),(106,80),(106,119),(107,40),(107,72),(107,136),(108,38),(108,61),(108,119),(109,76),(109,96),(109,131),(110,15),(110,93),(110,111),(111,67),(111,87),(111,108),(112,86),(112,92),(112,105),(112,107),(113,134),(113,140),(113,141),(114,133),(115,11),(115,132),(116,43),(116,139),(117,1),(118,143),(119,45),(119,126),(120,97),(120,109),(120,129),(121,122),(122,47),(122,134),(123,44),(124,46),(124,135),(125,133),(126,132),(127,135),(127,143),(128,49),(128,130),(129,48),(129,131),(130,125),(131,124),(131,127),(132,98),(134,128),(134,143),(135,123),(136,74),(136,121),(137,66),(138,83),(138,104),(138,116),(139,70),(139,117),(140,75),(140,91),(140,128),(141,75),(141,143),(142,59),(142,113),(142,122),(143,130)],144)
 => ? = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => [[1,1,1,1,1,2],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ([(0,2),(0,16),(0,17),(1,15),(1,39),(2,10),(2,14),(3,51),(3,137),(4,196),(5,50),(5,195),(6,202),(7,150),(8,46),(8,228),(9,32),(9,58),(10,27),(10,227),(11,28),(11,174),(12,33),(12,170),(13,36),(13,48),(13,223),(14,20),(14,47),(14,227),(15,26),(15,37),(15,224),(16,1),(16,136),(17,13),(17,31),(17,136),(18,219),(18,222),(19,163),(19,164),(20,166),(20,215),(21,213),(21,221),(22,53),(22,220),(22,232),(23,159),(23,212),(24,103),(24,160),(25,161),(25,162),(26,41),(26,225),(26,231),(27,168),(27,233),(28,153),(28,217),(29,158),(29,165),(30,154),(30,209),(31,155),(31,223),(32,156),(32,157),(33,100),(33,216),(34,101),(34,210),(35,52),(35,214),(35,218),(36,40),(36,167),(36,211),(37,225),(37,229),(38,102),(38,230),(39,168),(39,224),(40,114),(40,182),(41,173),(41,180),(42,131),(42,132),(42,152),(43,133),(43,134),(43,192),(44,97),(44,128),(44,199),(45,77),(45,135),(45,148),(46,78),(46,129),(46,198),(47,166),(47,193),(47,194),(48,130),(48,167),(48,194),(49,65),(49,99),(49,151),(50,62),(50,106),(50,107),(51,43),(51,175),(51,182),(51,229),(52,112),(52,113),(52,189),(53,116),(53,120),(53,171),(54,301),(54,317),(55,329),(56,308),(56,342),(57,295),(57,334),(58,12),(59,11),(60,253),(61,239),(61,244),(62,239),(62,250),(63,249),(63,338),(64,237),(65,5),(65,335),(66,278),(66,335),(67,246),(67,278),(68,281),(68,337),(69,277),(69,311),(70,274),(71,245),(71,277),(72,281),(72,312),(73,280),(73,314),(73,319),(74,247),(74,275),(74,313),(75,276),(75,334),(76,280),(76,315),(76,337),(77,247),(77,279),(78,249),(78,285),(79,259),(80,263),(80,269),(81,236),(81,262),(81,267),(82,235),(82,261),(82,267),(83,261),(83,266),(84,262),(84,266),(84,275),(85,329),(86,332),(87,305),(87,340),(88,304),(88,339),(89,341),(90,290),(91,270),(91,306),(92,238),(92,271),(93,238),(93,268),(94,273),(94,309),(95,269),(95,307),(96,237),(96,274),(97,248),(97,283),(98,264),(98,265),(99,30),(99,246),(99,335),(100,242),(101,252),(102,257),(103,25),(103,328),(104,132),(104,330),(105,152),(105,330),(106,118),(106,239),(107,140),(107,250),(108,54),(109,63),(109,293),(110,58),(111,100),(111,245),(112,109),(112,256),(113,138),(113,254),(114,110),(114,333),(115,164),(116,188),(116,325),(117,143),(118,94),(118,291),(119,174),(119,339),(120,169),(120,325),(121,78),(121,255),(122,147),(122,252),(123,60),(123,327),(124,183),(124,331),(125,70),(126,72),(127,69),(127,255),(128,191),(128,248),(129,208),(129,249),(130,165),(130,240),(131,87),(131,235),(131,236),(132,170),(132,235),(133,105),(133,243),(134,120),(134,243),(134,331),(135,44),(135,200),(135,247),(135,336),(136,3),(136,155),(137,4),(137,175),(138,129),(138,260),(139,54),(139,276),(139,292),(140,142),(140,272),(141,56),(141,272),(142,153),(142,273),(142,342),(143,57),(143,242),(143,271),(144,73),(144,263),(144,300),(145,73),(145,286),(145,297),(146,71),(146,282),(146,324),(147,69),(147,244),(147,282),(148,119),(148,279),(148,336),(149,76),(149,241),(149,270),(149,286),(150,19),(150,115),(151,23),(151,204),(151,246),(152,18),(152,177),(152,236),(153,90),(153,241),(154,59),(155,137),(156,111),(156,324),(157,89),(157,324),(158,9),(158,110),(159,121),(159,326),(160,55),(160,328),(161,125),(162,176),(163,79),(163,289),(164,70),(164,289),(165,34),(165,203),(166,21),(166,179),(166,251),(167,22),(167,178),(167,240),(168,181),(169,63),(169,206),(170,205),(170,216),(171,121),(171,127),(172,61),(172,62),(172,332),(173,60),(173,190),(174,126),(174,217),(175,133),(175,196),(175,333),(176,64),(176,96),(177,143),(177,219),(177,318),(178,187),(178,232),(179,187),(179,213),(180,102),(180,186),(181,123),(181,214),(182,124),(182,134),(182,333),(183,82),(183,83),(183,258),(184,81),(184,84),(184,256),(184,258),(185,127),(185,147),(185,326),(186,83),(186,84),(186,257),(187,122),(187,185),(188,71),(188,111),(189,74),(189,135),(189,254),(189,256),(190,74),(190,77),(190,253),(191,68),(191,76),(191,316),(192,112),(192,184),(192,331),(193,67),(193,151),(193,251),(194,178),(194,179),(195,59),(195,107),(196,42),(196,104),(196,105),(197,88),(197,119),(197,260),(198,56),(198,142),(198,285),(199,80),(199,144),(199,283),(200,199),(200,201),(200,303),(201,144),(201,145),(201,294),(202,55),(202,85),(203,210),(203,226),(204,172),(204,212),(204,302),(205,108),(205,139),(206,92),(206,93),(206,338),(207,149),(207,191),(207,287),(208,91),(208,149),(208,284),(209,138),(209,197),(210,146),(210,157),(210,252),(211,114),(211,158),(211,240),(212,61),(212,106),(212,326),(213,101),(213,122),(214,113),(214,209),(214,327),(215,65),(215,66),(215,251),(216,75),(216,139),(216,242),(217,68),(217,72),(217,241),(218,45),(218,189),(218,190),(218,327),(219,57),(219,75),(219,288),(220,116),(220,226),(221,86),(221,172),(222,80),(222,95),(222,288),(223,29),(223,130),(223,211),(224,35),(224,181),(224,231),(225,38),(225,180),(225,234),(226,146),(226,156),(226,188),(227,49),(227,193),(227,215),(227,233),(228,140),(228,141),(228,198),(229,124),(229,192),(229,234),(230,81),(230,82),(230,131),(230,257),(231,123),(231,173),(231,218),(232,159),(232,171),(232,185),(233,66),(233,67),(233,99),(234,183),(234,184),(234,186),(234,230),(235,205),(235,340),(236,222),(236,305),(236,318),(237,322),(238,320),(239,6),(239,291),(240,203),(240,220),(241,290),(241,312),(241,337),(242,292),(242,334),(243,325),(243,330),(244,291),(244,311),(245,103),(245,310),(246,8),(246,302),(247,97),(247,303),(247,346),(248,316),(249,284),(250,272),(251,204),(251,221),(251,278),(252,89),(252,282),(253,88),(253,279),(253,313),(254,260),(254,313),(254,336),(255,94),(255,285),(255,311),(256,200),(256,275),(256,293),(257,87),(257,261),(257,262),(258,206),(258,266),(258,267),(258,293),(259,322),(260,207),(260,208),(260,304),(261,340),(261,353),(262,305),(262,353),(263,98),(263,319),(263,352),(264,350),(265,96),(265,350),(266,93),(266,323),(266,353),(267,92),(267,318),(267,353),(268,297),(268,306),(269,352),(270,312),(270,315),(270,351),(271,292),(271,295),(271,320),(272,7),(272,342),(273,90),(273,354),(274,322),(275,303),(275,323),(276,98),(276,317),(276,349),(277,310),(278,86),(278,302),(279,339),(279,346),(280,299),(280,347),(281,348),(282,277),(282,341),(283,269),(283,300),(284,286),(284,306),(285,273),(285,308),(286,280),(286,290),(286,351),(287,270),(287,316),(288,263),(288,276),(288,295),(288,307),(289,259),(289,274),(290,298),(290,299),(291,202),(291,309),(292,301),(292,349),(293,201),(293,323),(293,338),(294,297),(294,300),(295,319),(295,321),(295,349),(296,307),(296,320),(297,314),(297,351),(298,345),(299,289),(299,345),(300,314),(300,352),(301,344),(302,228),(302,332),(303,283),(303,294),(304,91),(304,287),(305,95),(305,296),(306,351),(307,317),(307,321),(307,352),(308,354),(309,85),(309,354),(310,328),(311,308),(311,309),(312,298),(312,348),(313,304),(313,346),(314,343),(314,347),(315,347),(315,348),(316,281),(316,315),(317,264),(317,344),(318,271),(318,288),(318,296),(319,265),(319,343),(320,301),(320,321),(321,343),(321,344),(323,268),(323,294),(324,24),(324,245),(324,341),(325,117),(326,118),(326,244),(326,255),(327,148),(327,197),(327,253),(327,254),(328,161),(328,329),(329,125),(330,117),(330,177),(331,109),(331,169),(331,258),(332,141),(332,250),(333,104),(333,243),(334,162),(334,349),(335,154),(335,195),(336,128),(336,207),(336,346),(337,163),(337,299),(337,348),(338,145),(338,268),(338,284),(339,126),(340,108),(341,160),(341,310),(342,150),(342,354),(343,350),(344,64),(344,350),(345,259),(346,248),(346,287),(347,345),(348,79),(348,345),(349,176),(349,265),(349,344),(350,237),(351,298),(351,347),(352,264),(352,343),(353,238),(353,296),(354,115)],355)
 => ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [[1,1,1,1,1,2],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]]
 => ?
 => ? = 2 + 1

Description

The number of maximal elements of a poset.

Matching statistic: St001435
(load all 6 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 17%

Values

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

Description

The number of missing boxes in the first row.

The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St000068The number of minimal elements in a poset. St001720The minimal length of a chain of small intervals in a lattice. St000914The sum of the values of the Möbius function of a poset. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000007The number of saliances of the permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. 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. St001866The nesting alignments of a signed permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St000390The number of runs of ones in a binary word. St000805The number of peaks of the associated bargraph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000983The length of the longest alternating subword. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.