- FindStat

Your data matches 38 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001502
(load all 2 compositions to match this statistic)

Mapping

St001502: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths. The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.

Matching statistic: St001199

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%

Values

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

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001498
(load all 3 compositions to match this statistic)

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 40%

Values

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

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: St001933

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 40%

Values

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

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St001280

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 40%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000474
(load all 2 compositions to match this statistic)

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 40%

Values

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

Description

Dyson's crank of a partition. Let

$\lambda$ be a partition and let

$o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let

$\mu(\lambda)$ be the number of parts that are strictly larger than

$o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as

$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$

Matching statistic: St001571
(load all 2 compositions to match this statistic)

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%

Values

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

Description

The Cartan determinant of the integer partition. Let

$p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let

$A=K[x]/(x^t)$ be the finite dimensional algebra over the field

$K$ and

$M$ the direct sum of the indecomposable

$A$ -modules of vector space dimension

$p_i$ for each

$i$ . Then the Cartan determinant of

$p$ is the Cartan determinant of the endomorphism algebra of

$M$ over

$A$ . Explicitly, this is the determinant of the matrix

$\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$ , where

$\bar p$ is the set of distinct parts of the partition.

Matching statistic: St000993

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 40%

Values

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

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St001060
(load all 5 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 40%

Values

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

Description

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

Matching statistic: St000264
(load all 3 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000538The number of even inversions of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . 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

$eA$ . St000422The energy of a graph, if it is integral. St001845The number of join irreducibles minus the rank of a lattice. St001330The hat guessing number of a graph. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000089The absolute variation of a composition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word.