- FindStat

Your data matches 249 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001487
(load all 3 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 = 0 + 1
[[2],[]]
 => 1 = 0 + 1
[[1,1],[]]
 => 1 = 0 + 1
[[2,1],[1]]
 => 2 = 1 + 1
[[3],[]]
 => 1 = 0 + 1
[[2,1],[]]
 => 1 = 0 + 1
[[3,1],[1]]
 => 2 = 1 + 1
[[2,2],[1]]
 => 2 = 1 + 1
[[3,2],[2]]
 => 2 = 1 + 1
[[1,1,1],[]]
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => 2 = 1 + 1
[[2,1,1],[1]]
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => 3 = 2 + 1
[[4],[]]
 => 1 = 0 + 1
[[3,1],[]]
 => 1 = 0 + 1
[[4,1],[1]]
 => 2 = 1 + 1
[[2,2],[]]
 => 1 = 0 + 1
[[3,2],[1]]
 => 2 = 1 + 1
[[4,2],[2]]
 => 2 = 1 + 1
[[2,1,1],[]]
 => 1 = 0 + 1
[[3,2,1],[1,1]]
 => 2 = 1 + 1
[[3,1,1],[1]]
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => 3 = 2 + 1
[[3,3],[2]]
 => 2 = 1 + 1
[[4,3],[3]]
 => 2 = 1 + 1
[[2,2,1],[1]]
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => 3 = 2 + 1
[[3,2,1],[2]]
 => 2 = 1 + 1
[[4,3,1],[3,1]]
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => 2 = 1 + 1
[[3,3,2],[2,2]]
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => 3 = 2 + 1
[[4,3,2],[3,2]]
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => 4 = 3 + 1
[[5],[]]
 => 1 = 0 + 1
[[4,1],[]]
 => 1 = 0 + 1
[[5,1],[1]]
 => 2 = 1 + 1
[[3,2],[]]
 => 1 = 0 + 1
[[4,2],[1]]
 => 2 = 1 + 1
[[5,2],[2]]
 => 2 = 1 + 1
[[3,1,1],[]]
 => 1 = 0 + 1
[[4,2,1],[1,1]]
 => 2 = 1 + 1
[[4,1,1],[1]]
 => 2 = 1 + 1

Description

The number of inner corners of a skew 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
[[2],[]]
 => []
 => 0
[[1,1],[]]
 => []
 => 0
[[2,1],[1]]
 => [1]
 => 1
[[3],[]]
 => []
 => 0
[[2,1],[]]
 => []
 => 0
[[3,1],[1]]
 => [1]
 => 1
[[2,2],[1]]
 => [1]
 => 1
[[3,2],[2]]
 => [2]
 => 1
[[1,1,1],[]]
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => 1
[[2,1,1],[1]]
 => [1]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => 2
[[4],[]]
 => []
 => 0
[[3,1],[]]
 => []
 => 0
[[4,1],[1]]
 => [1]
 => 1
[[2,2],[]]
 => []
 => 0
[[3,2],[1]]
 => [1]
 => 1
[[4,2],[2]]
 => [2]
 => 1
[[2,1,1],[]]
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => 1
[[3,1,1],[1]]
 => [1]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => 2
[[3,3],[2]]
 => [2]
 => 1
[[4,3],[3]]
 => [3]
 => 1
[[2,2,1],[1]]
 => [1]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => 2
[[3,2,1],[2]]
 => [2]
 => 1
[[4,3,1],[3,1]]
 => [3,1]
 => 2
[[2,2,2],[1,1]]
 => [1,1]
 => 1
[[3,3,2],[2,2]]
 => [2,2]
 => 1
[[3,2,2],[2,1]]
 => [2,1]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => 2
[[1,1,1,1],[]]
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 2
[[2,1,1,1],[1]]
 => [1]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 2
[[3,2,1,1],[2,1]]
 => [2,1]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 3
[[5],[]]
 => []
 => 0
[[4,1],[]]
 => []
 => 0
[[5,1],[1]]
 => [1]
 => 1
[[3,2],[]]
 => []
 => 0
[[4,2],[1]]
 => [1]
 => 1
[[5,2],[2]]
 => [2]
 => 1
[[3,1,1],[]]
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => 1
[[4,1,1],[1]]
 => [1]
 => 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: 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 = 0 + 1
[[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => 2 = 1 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => ([],3)
 => 3 = 2 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => 4 = 3 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => 2 = 1 + 1
[[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1

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 = 0 + 1
[[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => 2 = 1 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2 = 1 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => ([],3)
 => 3 = 2 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2 = 1 + 1
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2 = 1 + 1
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => 3 = 2 + 1
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => 3 = 2 + 1
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => 4 = 3 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2 = 1 + 1
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => 2 = 1 + 1
[[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2 = 1 + 1
[[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2 = 1 + 1
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => 3 = 2 + 1
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1

Description

The number of maximal elements of a poset.

Matching statistic: St000318
(load all 3 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 = 0 + 1
[[2],[]]
 => []
 => 1 = 0 + 1
[[1,1],[]]
 => []
 => 1 = 0 + 1
[[2,1],[1]]
 => [1]
 => 2 = 1 + 1
[[3],[]]
 => []
 => 1 = 0 + 1
[[2,1],[]]
 => []
 => 1 = 0 + 1
[[3,1],[1]]
 => [1]
 => 2 = 1 + 1
[[2,2],[1]]
 => [1]
 => 2 = 1 + 1
[[3,2],[2]]
 => [2]
 => 2 = 1 + 1
[[1,1,1],[]]
 => []
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[[2,1,1],[1]]
 => [1]
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => [2,1]
 => 3 = 2 + 1
[[4],[]]
 => []
 => 1 = 0 + 1
[[3,1],[]]
 => []
 => 1 = 0 + 1
[[4,1],[1]]
 => [1]
 => 2 = 1 + 1
[[2,2],[]]
 => []
 => 1 = 0 + 1
[[3,2],[1]]
 => [1]
 => 2 = 1 + 1
[[4,2],[2]]
 => [2]
 => 2 = 1 + 1
[[2,1,1],[]]
 => []
 => 1 = 0 + 1
[[3,2,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[[3,1,1],[1]]
 => [1]
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => [2,1]
 => 3 = 2 + 1
[[3,3],[2]]
 => [2]
 => 2 = 1 + 1
[[4,3],[3]]
 => [3]
 => 2 = 1 + 1
[[2,2,1],[1]]
 => [1]
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => [2,1]
 => 3 = 2 + 1
[[3,2,1],[2]]
 => [2]
 => 2 = 1 + 1
[[4,3,1],[3,1]]
 => [3,1]
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[[3,3,2],[2,2]]
 => [2,2]
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => [2,1]
 => 3 = 2 + 1
[[4,3,2],[3,2]]
 => [3,2]
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => []
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => [1]
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 4 = 3 + 1
[[5],[]]
 => []
 => 1 = 0 + 1
[[4,1],[]]
 => []
 => 1 = 0 + 1
[[5,1],[1]]
 => [1]
 => 2 = 1 + 1
[[3,2],[]]
 => []
 => 1 = 0 + 1
[[4,2],[1]]
 => [1]
 => 2 = 1 + 1
[[5,2],[2]]
 => [2]
 => 2 = 1 + 1
[[3,1,1],[]]
 => []
 => 1 = 0 + 1
[[4,2,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[[4,1,1],[1]]
 => [1]
 => 2 = 1 + 1

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000292
(load all 9 compositions to match this statistic)

Mapping

Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => [1]
 => 10 => 0
[[2],[]]
 => [2]
 => 100 => 0
[[1,1],[]]
 => [1,1]
 => 110 => 0
[[2,1],[1]]
 => [2,1]
 => 1010 => 1
[[3],[]]
 => [3]
 => 1000 => 0
[[2,1],[]]
 => [2,1]
 => 1010 => 1
[[3,1],[1]]
 => [3,1]
 => 10010 => 1
[[2,2],[1]]
 => [2,2]
 => 1100 => 0
[[3,2],[2]]
 => [3,2]
 => 10100 => 1
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 1
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 2
[[4],[]]
 => [4]
 => 10000 => 0
[[3,1],[]]
 => [3,1]
 => 10010 => 1
[[4,1],[1]]
 => [4,1]
 => 100010 => 1
[[2,2],[]]
 => [2,2]
 => 1100 => 0
[[3,2],[1]]
 => [3,2]
 => 10100 => 1
[[4,2],[2]]
 => [4,2]
 => 100100 => 1
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 2
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 2
[[3,3],[2]]
 => [3,3]
 => 11000 => 0
[[4,3],[3]]
 => [4,3]
 => 101000 => 1
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 1
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 2
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 2
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 0
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 2
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 2
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 2
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 2
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 3
[[5],[]]
 => [5]
 => 100000 => 0
[[4,1],[]]
 => [4,1]
 => 100010 => 1
[[5,1],[1]]
 => [5,1]
 => 1000010 => 1
[[3,2],[]]
 => [3,2]
 => 10100 => 1
[[4,2],[1]]
 => [4,2]
 => 100100 => 1
[[5,2],[2]]
 => [5,2]
 => 1000100 => 1
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 2
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 1

Description

The number of ascents of a binary word.

Matching statistic: St000053
(load all 10 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 = 0 + 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1

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 = 0 + 1
[[2],[]]
 => [2]
 => 100 => 1 = 0 + 1
[[1,1],[]]
 => [1,1]
 => 110 => 1 = 0 + 1
[[2,1],[1]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3],[]]
 => [3]
 => 1000 => 1 = 0 + 1
[[2,1],[]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3,1],[1]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[2,2],[1]]
 => [2,2]
 => 1100 => 1 = 0 + 1
[[3,2],[2]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[4],[]]
 => [4]
 => 10000 => 1 = 0 + 1
[[3,1],[]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[4,1],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[2,2],[]]
 => [2,2]
 => 1100 => 1 = 0 + 1
[[3,2],[1]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[4,2],[2]]
 => [4,2]
 => 100100 => 2 = 1 + 1
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 2 = 1 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 3 = 2 + 1
[[3,3],[2]]
 => [3,3]
 => 11000 => 1 = 0 + 1
[[4,3],[3]]
 => [4,3]
 => 101000 => 2 = 1 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 2 = 1 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 2 = 1 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 3 = 2 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 4 = 3 + 1
[[5],[]]
 => [5]
 => 100000 => 1 = 0 + 1
[[4,1],[]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[5,1],[1]]
 => [5,1]
 => 1000010 => 2 = 1 + 1
[[3,2],[]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[4,2],[1]]
 => [4,2]
 => 100100 => 2 = 1 + 1
[[5,2],[2]]
 => [5,2]
 => 1000100 => 2 = 1 + 1
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 2 = 1 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 3 = 2 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 2 = 1 + 1

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 9 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 = 0 + 1
[[2],[]]
 => [2]
 => 100 => 1 = 0 + 1
[[1,1],[]]
 => [1,1]
 => 110 => 1 = 0 + 1
[[2,1],[1]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3],[]]
 => [3]
 => 1000 => 1 = 0 + 1
[[2,1],[]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3,1],[1]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[2,2],[1]]
 => [2,2]
 => 1100 => 1 = 0 + 1
[[3,2],[2]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[4],[]]
 => [4]
 => 10000 => 1 = 0 + 1
[[3,1],[]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[4,1],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[2,2],[]]
 => [2,2]
 => 1100 => 1 = 0 + 1
[[3,2],[1]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[4,2],[2]]
 => [4,2]
 => 100100 => 2 = 1 + 1
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 2 = 1 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 3 = 2 + 1
[[3,3],[2]]
 => [3,3]
 => 11000 => 1 = 0 + 1
[[4,3],[3]]
 => [4,3]
 => 101000 => 2 = 1 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 2 = 1 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 2 = 1 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 3 = 2 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 4 = 3 + 1
[[5],[]]
 => [5]
 => 100000 => 1 = 0 + 1
[[4,1],[]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[5,1],[1]]
 => [5,1]
 => 1000010 => 2 = 1 + 1
[[3,2],[]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[4,2],[1]]
 => [4,2]
 => 100100 => 2 = 1 + 1
[[5,2],[2]]
 => [5,2]
 => 1000100 => 2 = 1 + 1
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 2 = 1 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 3 = 2 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 2 = 1 + 1

Description

The number of runs of ones in a binary word.

Matching statistic: St001169
(load all 10 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 = 0 + 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1

Description

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

The following 239 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000015The number of peaks of a Dyck path. 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. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. 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. 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. 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. St000991The number of right-to-left minima 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. 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. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001424The number of distinct squares in a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000144The pyramid weight of the Dyck path. St000293The number of inversions of a binary word. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in 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. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001488The number of corners of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000761The number of ascents in an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000982The length of the longest constant subword. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001712The number of natural descents of a standard Young tableau. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001960The number of descents of a permutation minus one if its first entry is not one. St000702The number of weak deficiencies of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000288The number of ones in a binary word. St000335The difference of lower and upper interactions. St000444The length of the maximal rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001432The order dimension of the partition. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001840The number of descents of a set partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000929The constant term of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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

$eA$ . 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

$eA$ . 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

$eA$ . St001249Sum of the odd parts of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000782The indicator function of whether a given perfect matching is an L & P matching. St000454The largest eigenvalue of a graph if it is integral. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001435The number of missing boxes in the first row. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.