- FindStat

Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000259
(load all 10 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001093
(load all 3 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

Matching statistic: St001232

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%

Values

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

Description

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

Matching statistic: St000456

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 49%●distinct values known / distinct values provided: 33%

Values

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

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

Matching statistic: St001199
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 49%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => [[1]]
 => [[1]]
 => [1,0]
 => ? = 0 + 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[0,1],[1,0]]
 => [1,1,0,0]
 => ? = 1 + 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,0],[0,1]]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => 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]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 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]]
 => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 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]]
 => [[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]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 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]]
 => [[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]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[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]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[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]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[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]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[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],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 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],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 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],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 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],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 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],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 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],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 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],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 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],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 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],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 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],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,-1,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 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],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 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],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 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],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 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],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 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],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 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],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 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],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,-1,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001498
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 49%●distinct values known / distinct values provided: 33%

Values

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

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St001714

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic

$0$ are wide partitions.

Matching statistic: St001933

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%

Values

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

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St000929

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%

Values

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

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St000706

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%

Values

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

Description

The product of the factorials of the multiplicities of an integer partition.

The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000993The multiplicity of the largest part of an integer partition. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. 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

$eA$ .