- FindStat

Your data matches 84 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000031
(load all 3 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycles in the cycle decomposition of a permutation.

Matching statistic: St001088
(load all 7 compositions to match this statistic)

Mapping

St001088: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 10 - 1
[]
 => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 2 - 1

Description

Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

Matching statistic: St000444

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 1
[]
 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1

Description

The length of the maximal rise of a Dyck path.

Matching statistic: St000442
(load all 3 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 4 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 8 - 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 2
[]
 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 2

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000306
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 89%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000695
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000695: Set partitions ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => {{1,2}}
 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,6},{2,3},{4,5}}
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2,3},{4}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1,6},{2},{3,4,5}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2},{3,4}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,6},{2},{3,4},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,6},{5}}
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,8},{2,3},{4,5},{6,7}}
 => ? = 5 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5},{6}}
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,8,9},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,10,11},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 1
[]
 => [1] => [1,0]
 => {{1}}
 => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10,11}}
 => ? = 2 - 1

Description

The number of blocks in the first part of the atomic decomposition of a set partition. Let

$\pi=(b_1,\dots,b_k)$ be a set partition with

$k$ blocks, such that

$\min b_i < \min b_{i+1}$ . Then this statistic is the smallest number

$\ell$ such that the union of the first

$\ell$ blocks

$b_1\cup\dots\cup b_\ell$ is an interval

$\{1,\dots,m\}$ . The analogue for the decomposition of permutations is [[St000501]].

Matching statistic: St000925
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => {{1,2}}
 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,6},{2,3},{4,5}}
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2,3},{4}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1,6},{2},{3,4,5}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2},{3,4}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,6},{2},{3,4},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,6},{5}}
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,8},{2,3},{4,5},{6,7}}
 => ? = 5 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5},{6}}
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,8,9},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,10,11},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 1
[]
 => [1] => [1,0]
 => {{1}}
 => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10}}
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10,11}}
 => ? = 2 - 1

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St000585
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000585: Set partitions ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => {{1,2}}
 => 0 = 2 - 2
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0 = 2 - 2
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0 = 2 - 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0 = 2 - 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,6},{2,3},{4,5}}
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2,3},{4}}
 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1,6},{2},{3,4,5}}
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2},{3,4}}
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,6},{2},{3,4},{5}}
 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,6},{5}}
 => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1,6},{2,3,4,5}}
 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,5,6},{2,3,4}}
 => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,8},{2,3},{4,5},{6,7}}
 => ? = 5 - 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 3 - 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5},{6}}
 => ? = 7 - 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 6 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,8,9},{2},{3},{4},{5},{6},{7}}
 => ? = 8 - 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9}}
 => ? = 2 - 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 9 - 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,10,11},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 2
[]
 => [1] => [1,0]
 => {{1}}
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10}}
 => ? = 2 - 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 10 - 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8,9,10,11}}
 => ? = 2 - 2

Description

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.

Matching statistic: St001004
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => [2,1] => 2
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 3
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 3
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 4
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 3
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => 5
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 5
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 4
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,7,6,5,4,1] => ? = 4
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 6
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,1,8,2,6,7,4,5] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,2,8,7,6,5,4,1] => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [4,3,2,8,7,6,5,1] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [2,3,8,1,7,4,5,6] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [2,3,8,5,7,1,4,6] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [2,8,4,5,6,1,3,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,8,7,6,5,4,3,1] => ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,8,7,1] => ? = 7
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,5,8,7,6,1] => ? = 6
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [2,3,8,5,6,7,1,4] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,9,8,1] => ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,8,10,9,1] => ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,8,9,11,10,1] => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [11,10,9,8,7,6,5,4,3,2,1] => ? = 2

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St000996
(load all 11 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,7,6,5,4,1] => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,1,8,2,6,7,4,5] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,2,8,7,6,5,4,1] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [4,3,2,8,7,6,5,1] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [2,3,8,1,7,4,5,6] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4 - 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [2,3,8,5,7,1,4,6] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4 - 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [2,8,4,5,6,1,3,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,8,7,6,5,4,3,1] => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,8,7,1] => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,5,8,7,6,1] => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [2,3,8,5,6,7,1,4] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => ? = 4 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,9,8,1] => ? = 8 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,8,10,9,1] => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,8,9,11,10,1] => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [11,10,9,8,7,6,5,4,3,2,1] => ? = 2 - 1

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000497The lcb statistic of a set partition. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000374The number of exclusive right-to-left minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000203The number of external nodes of a binary tree. St000007The number of saliances of the permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St000053The number of valleys of the Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000991The number of right-to-left minima of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000711The number of big exceedences of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000236The number of cyclical small weak excedances. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000806The semiperimeter of the associated bargraph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001889The size of the connectivity set of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000383The last part of an integer composition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000710The number of big deficiencies of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000942The number of critical left to right maxima of the parking functions. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000894The trace of an alternating sign matrix. St001330The hat guessing number of a graph. St000075The orbit size of a standard tableau under promotion. St000907The number of maximal antichains of minimal length in a poset. St000441The number of successions of a permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000054The first entry of the permutation. St000352The Elizalde-Pak rank of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.