Your data matches 151 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001227
(load all 47 compositions to match this statistic)
Mapping
St001227: 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]
 => 1
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000010
(load all 9 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => []
 => 0
[1,0,1,0]
 => [1]
 => 1
[1,1,0,0]
 => []
 => 0
[1,0,1,0,1,0]
 => [2,1]
 => 2
[1,0,1,1,0,0]
 => [1,1]
 => 2
[1,1,0,0,1,0]
 => [2]
 => 1
[1,1,0,1,0,0]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
Description
The length of the partition.
Matching statistic: St001291
(load all 70 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000019
(load all 57 compositions to match this statistic)
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => 0
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
Description
The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.
Matching statistic: St000051
(load all 5 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [[.,.],[.,.]]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => 3
Description
The size of the left subtree of a binary tree.
Matching statistic: St000067
(load all 6 compositions to match this statistic)
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1]]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 3
Description
The inversion number of the alternating sign matrix. If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as $$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$ When restricted to permutation matrices, this gives the usual inversion number of the permutation.
Matching statistic: St000141
(load all 26 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,2,3,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,3,4,2,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,3,4,1,2] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,2,4,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,2,3,4,1] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,2,3,1,4] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,5,3,2,1] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,5,2,3,1] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,5,2,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,3,5,1,2] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,2,3,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 3
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
(load all 10 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => []
 => []
 => 0
[1,0,1,0]
 => [1]
 => [1]
 => 1
[1,1,0,0]
 => []
 => []
 => 0
[1,0,1,0,1,0]
 => [2,1]
 => [2,1]
 => 2
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => 1
[1,1,0,1,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,2,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [3,1,1]
 => 3
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [3,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [3]
 => 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [2,1,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [2,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [4,3,2,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [4,3,2]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [4,3,1,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [4,3,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [4,3]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [4,2,2,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [4,2,2]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [4,2,1,1]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [4,2,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [4,2]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [4,1,1]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [4]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,3,2,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,3,2]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [3,3,1,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [3,3,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [3,3]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,2,2,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,2,2]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [3,2,1,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [3,1,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [3,1,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [3,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [3]
 => 3
Description
The largest part of an integer partition.
Matching statistic: St000204
(load all 5 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[.,[[.,.],[[.,.],.]]],.]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[.,[[.,[.,.]],[.,.]]],.]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[.,[[[.,.],[.,.]],.]],.]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[.,[[[.,[.,.]],.],.]],.]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[[.,.],[.,[[.,.],.]]],.]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[[.,.],[[.,[.,.]],.]],.]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[[.,.],[[[.,.],.],.]],.]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[[.,[.,.]],[[.,.],.]],.]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[[.,[[.,.],.]],[.,.]],.]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [[[.,[[.,.],[.,.]]],.],.]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [[[.,[[.,[.,.]],.]],.],.]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => 3
Description
The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000304
(load all 5 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [2,5,4,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,5,4,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [2,5,4,3,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,5,4,3,1] => 3
Description
The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St000356The number of occurrences of the pattern 13-2. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000240The number of indices that are not small excedances. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000839The largest opener of a set partition. St001497The position of the largest weak excedence of a permutation. St000009The charge of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000030The sum of the descent differences of a permutations. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000305The inverse major index of a permutation. 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. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000703The number of deficiencies of a permutation. St000845The maximal number of elements covered by an element in a poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001481The minimal height of a peak of a Dyck path. St001725The harmonious chromatic number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000844The size of the largest block in the direct sum decomposition of a permutation. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000676The number of odd rises of a Dyck path. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001280The number of parts of an integer partition that are at least two. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001777The number of weak descents in an integer composition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000006The dinv of a Dyck path. St000015The number of peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001462The number of factors of a standard tableaux under concatenation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000840The number of closers smaller than the largest opener in a perfect matching. St000105The number of blocks in the set partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000925The number of topologically connected components of a set partition. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001152The number of pairs with even minimum in a perfect matching. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001645The pebbling number of a connected graph. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001434The number of negative sum pairs of a signed permutation. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001861The number of Bruhat lower covers of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000041The number of nestings of a perfect matching. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000005The bounce statistic of a Dyck path. St000358The number of occurrences of the pattern 31-2. St001209The pmaj statistic of a parking function. St001811The Castelnuovo-Mumford regularity of a permutation. St000115The single entry in the last row. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2.