- FindStat

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

Matching statistic: St001487
(load all 2 compositions to match this statistic)

Mapping

St001487: Skew 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]]
 => 2
[[3],[]]
 => 1
[[2,1],[]]
 => 1
[[3,1],[1]]
 => 2
[[2,2],[1]]
 => 2
[[3,2],[2]]
 => 2
[[1,1,1],[]]
 => 1
[[2,2,1],[1,1]]
 => 2
[[2,1,1],[1]]
 => 2
[[3,2,1],[2,1]]
 => 3
[[4],[]]
 => 1
[[3,1],[]]
 => 1
[[4,1],[1]]
 => 2
[[2,2],[]]
 => 1
[[3,2],[1]]
 => 2
[[4,2],[2]]
 => 2
[[2,1,1],[]]
 => 1
[[3,2,1],[1,1]]
 => 2
[[3,1,1],[1]]
 => 2
[[4,2,1],[2,1]]
 => 3
[[3,3],[2]]
 => 2
[[4,3],[3]]
 => 2
[[2,2,1],[1]]
 => 2
[[3,3,1],[2,1]]
 => 3
[[3,2,1],[2]]
 => 2
[[4,3,1],[3,1]]
 => 3
[[2,2,2],[1,1]]
 => 2
[[3,3,2],[2,2]]
 => 2
[[3,2,2],[2,1]]
 => 3
[[4,3,2],[3,2]]
 => 3
[[1,1,1,1],[]]
 => 1
[[2,2,2,1],[1,1,1]]
 => 2
[[2,2,1,1],[1,1]]
 => 2
[[3,3,2,1],[2,2,1]]
 => 3
[[2,1,1,1],[1]]
 => 2
[[3,2,2,1],[2,1,1]]
 => 3
[[3,2,1,1],[2,1]]
 => 3
[[4,3,2,1],[3,2,1]]
 => 4
[[5],[]]
 => 1
[[4,1],[]]
 => 1
[[5,1],[1]]
 => 2
[[3,2],[]]
 => 1
[[4,2],[1]]
 => 2
[[5,2],[2]]
 => 2
[[3,1,1],[]]
 => 1
[[4,2,1],[1,1]]
 => 2
[[4,1,1],[1]]
 => 2

Description

The number of inner corners of a skew partition.

Matching statistic: St000068

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
St000068: 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)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2
[[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)
 => 1
[[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)
 => 1
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => 2
[[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)
 => 2
[[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)
 => 3
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => 2
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => 3
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => 2
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => 3
[[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)
 => 1
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => 2
[[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1
[[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)
 => 1
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => 2
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2

Description

The number of minimal elements in a poset.

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: St000159
(load all 2 compositions to match this statistic)

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ 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 = 2 - 1
[[3],[]]
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4],[]]
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 3 = 4 - 1
[[5],[]]
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[4,1,1],[1]]
 => [1]
 => 1 = 2 - 1

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the 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: St000291
(load all 6 compositions to match this statistic)

Mapping

Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => [1]
 => 10 => 1
[[2],[]]
 => [2]
 => 100 => 1
[[1,1],[]]
 => [1,1]
 => 110 => 1
[[2,1],[1]]
 => [2,1]
 => 1010 => 2
[[3],[]]
 => [3]
 => 1000 => 1
[[2,1],[]]
 => [2,1]
 => 1010 => 2
[[3,1],[1]]
 => [3,1]
 => 10010 => 2
[[2,2],[1]]
 => [2,2]
 => 1100 => 1
[[3,2],[2]]
 => [3,2]
 => 10100 => 2
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 3
[[4],[]]
 => [4]
 => 10000 => 1
[[3,1],[]]
 => [3,1]
 => 10010 => 2
[[4,1],[1]]
 => [4,1]
 => 100010 => 2
[[2,2],[]]
 => [2,2]
 => 1100 => 1
[[3,2],[1]]
 => [3,2]
 => 10100 => 2
[[4,2],[2]]
 => [4,2]
 => 100100 => 2
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 3
[[3,3],[2]]
 => [3,3]
 => 11000 => 1
[[4,3],[3]]
 => [4,3]
 => 101000 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 4
[[5],[]]
 => [5]
 => 100000 => 1
[[4,1],[]]
 => [4,1]
 => 100010 => 2
[[5,1],[1]]
 => [5,1]
 => 1000010 => 2
[[3,2],[]]
 => [3,2]
 => 10100 => 2
[[4,2],[1]]
 => [4,2]
 => 100100 => 2
[[5,2],[2]]
 => [5,2]
 => 1000100 => 2
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 2

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 6 compositions to match this statistic)

Mapping

Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => [1]
 => 10 => 1
[[2],[]]
 => [2]
 => 100 => 1
[[1,1],[]]
 => [1,1]
 => 110 => 1
[[2,1],[1]]
 => [2,1]
 => 1010 => 2
[[3],[]]
 => [3]
 => 1000 => 1
[[2,1],[]]
 => [2,1]
 => 1010 => 2
[[3,1],[1]]
 => [3,1]
 => 10010 => 2
[[2,2],[1]]
 => [2,2]
 => 1100 => 1
[[3,2],[2]]
 => [3,2]
 => 10100 => 2
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 2
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 3
[[4],[]]
 => [4]
 => 10000 => 1
[[3,1],[]]
 => [3,1]
 => 10010 => 2
[[4,1],[1]]
 => [4,1]
 => 100010 => 2
[[2,2],[]]
 => [2,2]
 => 1100 => 1
[[3,2],[1]]
 => [3,2]
 => 10100 => 2
[[4,2],[2]]
 => [4,2]
 => 100100 => 2
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 3
[[3,3],[2]]
 => [3,3]
 => 11000 => 1
[[4,3],[3]]
 => [4,3]
 => 101000 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 2
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 3
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 4
[[5],[]]
 => [5]
 => 100000 => 1
[[4,1],[]]
 => [4,1]
 => 100010 => 2
[[5,1],[1]]
 => [5,1]
 => 1000010 => 2
[[3,2],[]]
 => [3,2]
 => 10100 => 2
[[4,2],[1]]
 => [4,2]
 => 100100 => 2
[[5,2],[2]]
 => [5,2]
 => 1000100 => 2
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 3
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 2

Description

The number of runs of ones in a binary word.

Matching statistic: St001169
(load all 6 compositions to match this statistic)

Mapping

Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001169: 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

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

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.

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

St000015The number of peaks of a Dyck path. St000292The number of ascents of a binary word. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. 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. St000672The number of minimal elements in Bruhat order not less than the permutation. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. 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. St000742The number of big ascents of a permutation after prepending zero. St000925The number of topologically connected components of a set partition. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St000386The number of factors DDU in a Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001566The length of the longest arithmetic progression in a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000646The number of big ascents of a permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000647The number of big descents of a permutation. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000488The number of cycles of a permutation of length at most 2. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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. St000455The second largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001435The number of missing boxes in the first row. St000264The girth of a graph, which is not a tree. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001712The number of natural descents of a standard Young tableau. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St001330The hat guessing number of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition.