- FindStat

Your data matches 147 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001068
(load all 6 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St001088
(load all 10 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000052
(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
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000053
(load all 6 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St001506
(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
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000011

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000013
(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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000105
(load all 3 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]]

$S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of

$\{ 1,\ldots,n\}$ into

$k$ blocks, see [1].

Matching statistic: St000167

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaves of an ordered tree. This is the number of nodes which do not have any children.

The following 137 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. 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: St000024The number of double up and double down steps of a Dyck path. St000065The number of entries equal to -1 in an alternating sign matrix. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000444The length of the maximal rise of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000925The number of topologically connected components of a set partition. 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. St000442The maximal area to the right of an up step of a Dyck path. St000497The lcb statistic of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000069The number of maximal elements of a poset. St000159The number of distinct parts of the integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000711The number of big exceedences of a permutation. St000007The number of saliances of the permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000546The number of global descents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. 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

$eA$ . St001489The maximum of the number of descents and the number of inverse descents. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000061The number of nodes on the left branch of a binary tree. St000619The number of cyclic descents of a permutation. St000654The first descent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000068The number of minimal elements in a poset. St000806The semiperimeter of the associated bargraph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. 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. St001875The number of simple modules with projective dimension at most 1. St000383The last part of an integer composition. St000702The number of weak deficiencies of a permutation. St000989The number of final rises of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. 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

$eA$ . St000075The orbit size of a standard tableau under promotion. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001867The number of alignments of type EN of a signed permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the 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. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000366The number of double descents of a permutation. St001712The number of natural descents of a standard Young tableau. St001866The nesting alignments of a signed permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000056The decomposition (or block) number of a permutation. St000492The rob statistic of a set partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001479The number of bridges of a graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001826The maximal number of leaves on a vertex of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000234The number of global ascents of a permutation. St000839The largest opener of a set partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001728The number of invisible descents of a permutation. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000230Sum of the minimal elements of the blocks of a set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001644The dimension of a graph.