- FindStat

Your data matches 219 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001001
(load all 12 compositions to match this statistic)

Mapping

St001001: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001181
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 84%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 3
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? ∊ {0,0,1,2,2,10}

Description

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001264
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 3
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,0,10}

Description

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

Matching statistic: St001292
(load all 5 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 3
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,10}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,0,10}

Description

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here

$A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].

Matching statistic: St001810
(load all 2 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => []
 => []
 => [] => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1] => 0
[1,1,0,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1] => 0
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 3
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [2,3,1] => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1] => 0
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [2,5,1,3,7,4,6] => ? ∊ {0,0,0,0,2,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,2,6,3,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,7,5,6] => ? ∊ {0,0,0,0,2,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,1,6,7,3,4] => ? ∊ {0,0,0,0,2,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,1,5,6,2,3] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,7,4,5] => ? ∊ {0,0,0,0,2,10}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,5,6,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,6,7,1,5] => ? ∊ {0,0,0,0,2,10}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,5,6,1,4] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [3,1,2] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? ∊ {0,0,0,0,2,10}

Description

The number of fixed points of a permutation smaller than its largest moved point.

Matching statistic: St000801
(load all 2 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 83%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => []
 => []
 => [] => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,1}
[1,1,0,0]
 => []
 => []
 => [] => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,3}
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? ∊ {0,3}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {1,6}
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? ∊ {1,6}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,6,3,5,4,7,2] => ? ∊ {0,0,0,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,4,3,6,1] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,5,3,4,6,7,2] => ? ∊ {0,0,0,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,6,3,5,4,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,0,0,10}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? ∊ {0,0,0,10}

Description

The number of occurrences of the vincular pattern |312 in a permutation. This is the number of occurrences of the pattern

$(3,1,2)$ , such that the letter matched by

$3$ is the first entry of the permutation.

Matching statistic: St001141
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 83%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,1}
[1,1,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,3}
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,3}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,6}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,6}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,2,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {0,2,2,10}
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,2,2,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,2,2,10}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,2,2,10}

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

Matching statistic: St000791
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 80%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,1}
[1,1,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,3}
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,3}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {1,6}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {1,6}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,2,2,10}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,2,10}

Description

The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].

Matching statistic: St000980
(load all 5 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 80%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,1}
[1,1,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,3}
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,3}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {1,6}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {1,6}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,2,2,10}
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,2,2,10}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,2,10}

Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path

$111011010000$ has three peaks in positions

$03, 15, 26$ . The boxes below

$03$ are

$01,02,\textbf{12}$ , the boxes below

$15$ are

$\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$ , and the boxes below

$26$ are

$\textbf{23},\textbf{24},25,\textbf{34},35,45$ . We thus obtain the four boxes in positions

$12,23,24,34$ that are below at least two peaks.

Matching statistic: St000516
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 78%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => []
 => []
 => [] => ? = 0
[1,0,1,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,1}
[1,1,0,0]
 => []
 => []
 => [] => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,3}
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? ∊ {0,3}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,1,3] => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {1,6}
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? ∊ {1,6}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [6,3,4,2,5,7,1] => ? ∊ {0,0,0,0,0,1,10}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => ? ∊ {0,0,0,0,0,1,10}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,4,7,1] => ? ∊ {0,0,0,0,0,1,10}
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => ? ∊ {0,0,0,0,0,1,10}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => ? ∊ {0,0,0,0,0,1,10}
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,1,3] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => [3,2,1] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0]
 => [1] => ? ∊ {0,0,0,0,0,1,10}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? ∊ {0,0,0,0,0,1,10}

Description

The number of stretching pairs of a permutation. This is the number of pairs

$(i,j)$ with

$\pi(i) < i < j < \pi(j)$ .

The following 209 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St000233The number of nestings of a set partition. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000376The bounce deficit of a Dyck path. St000664The number of right ropes of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001811The Castelnuovo-Mumford regularity of a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000732The number of double deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001130The number of two successive successions in a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001846The number of elements which do not have a complement in the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000348The non-inversion sum of a binary word. St000377The dinv defect of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000995The largest even part of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001910The height of the middle non-run of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000741The Colin de Verdière graph invariant. St000669The number of permutations obtained by switching ascents or descents of size 2. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001820The size of the image of the pop stack sorting operator. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St000369The dinv deficit of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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

$eA$ . St001330The hat guessing number of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001052The length of the exterior of a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000100The number of linear extensions of a poset. St000694The number of affine bounded permutations that project to a given permutation. St000907The number of maximal antichains of minimal length in a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph.