Your data matches 102 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000069
(load all 3 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => 1
[[2],[]]
 => ([(0,1)],2)
 => 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1
[[2,1],[1]]
 => ([],2)
 => 2
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2
[[3,1],[1]]
 => ([(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 2
[[3,2,1],[2,1]]
 => ([],3)
 => 3
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => 2
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => 3
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => 2
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => 3
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => 2
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => 2
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => 3
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => 3
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => 2
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => 2
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => 3
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => 2
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => 2
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => 3
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => 3
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => 3
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => 4
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => 2
[[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => 3
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2
Description
The number of maximal elements of a poset.
Matching statistic: St000318
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => 1
[[2],[]]
 => []
 => 1
[[1,1],[]]
 => []
 => 1
[[2,1],[1]]
 => [1]
 => 2
[[3],[]]
 => []
 => 1
[[2,1],[]]
 => []
 => 1
[[3,1],[1]]
 => [1]
 => 2
[[2,2],[1]]
 => [1]
 => 2
[[3,2],[2]]
 => [2]
 => 2
[[1,1,1],[]]
 => []
 => 1
[[2,2,1],[1,1]]
 => [1,1]
 => 2
[[2,1,1],[1]]
 => [1]
 => 2
[[3,2,1],[2,1]]
 => [2,1]
 => 3
[[4],[]]
 => []
 => 1
[[3,1],[]]
 => []
 => 1
[[4,1],[1]]
 => [1]
 => 2
[[2,2],[]]
 => []
 => 1
[[3,2],[1]]
 => [1]
 => 2
[[4,2],[2]]
 => [2]
 => 2
[[2,1,1],[]]
 => []
 => 1
[[3,2,1],[1,1]]
 => [1,1]
 => 2
[[3,1,1],[1]]
 => [1]
 => 2
[[4,2,1],[2,1]]
 => [2,1]
 => 3
[[3,3],[2]]
 => [2]
 => 2
[[4,3],[3]]
 => [3]
 => 2
[[2,2,1],[1]]
 => [1]
 => 2
[[3,3,1],[2,1]]
 => [2,1]
 => 3
[[3,2,1],[2]]
 => [2]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => 3
[[2,2,2],[1,1]]
 => [1,1]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => 3
[[1,1,1,1],[]]
 => []
 => 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 2
[[2,2,1,1],[1,1]]
 => [1,1]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 3
[[2,1,1,1],[1]]
 => [1]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 4
[[5],[]]
 => []
 => 1
[[4,1],[]]
 => []
 => 1
[[5,1],[1]]
 => [1]
 => 2
[[3,2],[]]
 => []
 => 1
[[4,2],[1]]
 => [1]
 => 2
[[5,2],[2]]
 => [2]
 => 2
[[3,1,1],[]]
 => []
 => 1
[[4,2,1],[1,1]]
 => [1,1]
 => 2
[[4,1,1],[1]]
 => [1]
 => 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000053
(load all 6 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001499
(load all 7 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001068
(load all 10 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000024
(load all 9 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000340
(load all 3 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 4
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 2
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1 = 2 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000052
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => []
 => [1,0]
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000105
(load all 2 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 2 = 1 + 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 1 + 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2 = 1 + 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3 = 2 + 1
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 1 + 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3 = 2 + 1
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3 = 2 + 1
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 2 + 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 1 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3 = 2 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3 = 2 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4 = 3 + 1
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2 = 1 + 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3 = 2 + 1
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 3 = 2 + 1
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 2 + 1
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 3 = 2 + 1
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3 = 2 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4 = 3 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3 = 2 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4 = 3 + 1
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3 = 2 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 3 = 2 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4 = 3 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 4 = 3 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2 = 1 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 3 = 2 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 3 = 2 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 4 = 3 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 3 = 2 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 4 = 3 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 4 = 3 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5 = 4 + 1
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 2 = 1 + 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 3 = 2 + 1
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => {{1,2,3,5},{4},{6}}
 => 3 = 2 + 1
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 2 + 1
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 3 = 2 + 1
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => {{1,2,5},{3,4},{6}}
 => 3 = 2 + 1
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3 = 2 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4 = 3 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 3 = 2 + 1
Description
The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000925The number of topologically connected components of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000159The number of distinct parts of the integer partition. St000068The number of minimal elements in a poset. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000991The number of right-to-left minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000522The number of 1-protected nodes of a rooted tree. St000702The number of weak deficiencies of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St000619The number of cyclic descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000386The number of factors DDU in a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. 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 $eA$. St001083The number of boxed occurrences of 132 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000646The number of big ascents of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000647The number of big descents of a permutation. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001487The number of inner corners of a skew partition. St001875The number of simple modules with projective dimension at most 1. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001840The number of descents of a set partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001557The number of inversions of the second entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000454The largest eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001435The number of missing boxes in the first row. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001712The number of natural descents of a standard Young tableau. St001621The number of atoms of a lattice.