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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St001487
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(load all 3 compositions to match this statistic)
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
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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: St000318
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
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: St000069
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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)
=> 1
[[2],[]]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> [[2,1],[1]]
=> ([],2)
=> 2
[[3],[]]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,1],[]]
=> [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> ([(1,2)],3)
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[[3,2],[2]]
=> [[3,1],[1]]
=> ([(1,2)],3)
=> 2
[[1,1,1],[]]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> ([(1,2)],3)
=> 2
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> ([(1,2)],3)
=> 2
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> ([],3)
=> 3
[[4],[]]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,1],[]]
=> [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,1],[1]]
=> [[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2],[]]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,2],[1]]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[[4,2],[2]]
=> [[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> ([(2,3)],4)
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[4,3],[3]]
=> [[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> 3
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> ([(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> 2
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> 3
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> ([(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> 2
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> 3
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> 2
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> 3
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> 3
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> ([],4)
=> 4
[[5],[]]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[4,1],[]]
=> [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[[5,1],[1]]
=> [[5,4],[4]]
=> ([(1,4),(3,2),(4,3)],5)
=> 2
[[3,2],[]]
=> [[3,3],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> ([(0,3),(1,4),(4,2)],5)
=> 2
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> ([(0,3),(1,4),(4,2)],5)
=> 2
Description
The number of maximal elements of a poset.
Matching statistic: St000015
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ 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]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3],[]]
=> []
=> []
=> [1,0]
=> 1
[[2,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[4],[]]
=> []
=> []
=> [1,0]
=> 1
[[3,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2],[]]
=> []
=> []
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[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]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[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]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[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]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> [1,0]
=> 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]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[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]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[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]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[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]
=> 4
[[5],[]]
=> []
=> []
=> [1,0]
=> 1
[[4,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2],[]]
=> []
=> []
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> [1,0]
=> 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000053
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00189: Skew partitions —rotate⟶ Skew partitions
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%
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]
=> [1,0,1,0]
=> 1
[[2],[]]
=> [[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[1]]
=> [[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3],[]]
=> [[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,2],[2]]
=> [[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4],[]]
=> [[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[4,1],[1]]
=> [[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,2],[1]]
=> [[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2],[2]]
=> [[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[4,3],[3]]
=> [[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[1,1,1,1],[]]
=> [[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]]
=> [[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [[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]]
=> [[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[3,2,2,1],[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
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[[4,3,2,1],[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
[[5],[]]
=> [[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[5,1],[1]]
=> [[5,4],[4]]
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000291
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00189: Skew partitions —rotate⟶ Skew partitions
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%
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],[]]
=> [1]
=> 10 => 1
[[2],[]]
=> [[2],[]]
=> [2]
=> 100 => 1
[[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 110 => 1
[[2,1],[1]]
=> [[2,1],[1]]
=> [2,1]
=> 1010 => 2
[[3],[]]
=> [[3],[]]
=> [3]
=> 1000 => 1
[[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 1100 => 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [3,2]
=> 10100 => 2
[[2,2],[1]]
=> [[2,1],[]]
=> [2,1]
=> 1010 => 2
[[3,2],[2]]
=> [[3,1],[1]]
=> [3,1]
=> 10010 => 2
[[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1110 => 1
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [2,1,1]
=> 10110 => 2
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [2,2,1]
=> 11010 => 2
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [3,2,1]
=> 101010 => 3
[[4],[]]
=> [[4],[]]
=> [4]
=> 10000 => 1
[[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 11000 => 1
[[4,1],[1]]
=> [[4,3],[3]]
=> [4,3]
=> 101000 => 2
[[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 1100 => 1
[[3,2],[1]]
=> [[3,2],[1]]
=> [3,2]
=> 10100 => 2
[[4,2],[2]]
=> [[4,2],[2]]
=> [4,2]
=> 100100 => 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 11100 => 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [3,2,2]
=> 101100 => 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [3,3,2]
=> 110100 => 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [4,3,2]
=> 1010100 => 3
[[3,3],[2]]
=> [[3,1],[]]
=> [3,1]
=> 10010 => 2
[[4,3],[3]]
=> [[4,1],[1]]
=> [4,1]
=> 100010 => 2
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 11010 => 2
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [3,2,1]
=> 101010 => 3
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [3,3,1]
=> 110010 => 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [4,3,1]
=> 1010010 => 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> [2,1,1]
=> 10110 => 2
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [3,1,1]
=> 100110 => 2
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [3,2,1]
=> 101010 => 3
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [4,2,1]
=> 1001010 => 3
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 1
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [2,1,1,1]
=> 101110 => 2
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> 110110 => 2
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> 1010110 => 3
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> 111010 => 2
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> 1011010 => 3
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> 1101010 => 3
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> 10101010 => 4
[[5],[]]
=> [[5],[]]
=> [5]
=> 100000 => 1
[[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 110000 => 1
[[5,1],[1]]
=> [[5,4],[4]]
=> [5,4]
=> 1010000 => 2
[[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 11000 => 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [4,3]
=> 101000 => 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [5,3]
=> 1001000 => 2
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 111000 => 1
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [4,3,3]
=> 1011000 => 2
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [4,4,3]
=> 1101000 => 2
Description
The number of descents of a binary word.
Matching statistic: St000390
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00189: Skew partitions —rotate⟶ Skew partitions
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%
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],[]]
=> [1]
=> 10 => 1
[[2],[]]
=> [[2],[]]
=> [2]
=> 100 => 1
[[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 110 => 1
[[2,1],[1]]
=> [[2,1],[1]]
=> [2,1]
=> 1010 => 2
[[3],[]]
=> [[3],[]]
=> [3]
=> 1000 => 1
[[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 1100 => 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [3,2]
=> 10100 => 2
[[2,2],[1]]
=> [[2,1],[]]
=> [2,1]
=> 1010 => 2
[[3,2],[2]]
=> [[3,1],[1]]
=> [3,1]
=> 10010 => 2
[[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1110 => 1
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [2,1,1]
=> 10110 => 2
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [2,2,1]
=> 11010 => 2
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [3,2,1]
=> 101010 => 3
[[4],[]]
=> [[4],[]]
=> [4]
=> 10000 => 1
[[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 11000 => 1
[[4,1],[1]]
=> [[4,3],[3]]
=> [4,3]
=> 101000 => 2
[[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 1100 => 1
[[3,2],[1]]
=> [[3,2],[1]]
=> [3,2]
=> 10100 => 2
[[4,2],[2]]
=> [[4,2],[2]]
=> [4,2]
=> 100100 => 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 11100 => 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [3,2,2]
=> 101100 => 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [3,3,2]
=> 110100 => 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [4,3,2]
=> 1010100 => 3
[[3,3],[2]]
=> [[3,1],[]]
=> [3,1]
=> 10010 => 2
[[4,3],[3]]
=> [[4,1],[1]]
=> [4,1]
=> 100010 => 2
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 11010 => 2
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [3,2,1]
=> 101010 => 3
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [3,3,1]
=> 110010 => 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [4,3,1]
=> 1010010 => 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> [2,1,1]
=> 10110 => 2
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [3,1,1]
=> 100110 => 2
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [3,2,1]
=> 101010 => 3
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [4,2,1]
=> 1001010 => 3
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 1
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [2,1,1,1]
=> 101110 => 2
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> 110110 => 2
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> 1010110 => 3
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> 111010 => 2
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> 1011010 => 3
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> 1101010 => 3
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> 10101010 => 4
[[5],[]]
=> [[5],[]]
=> [5]
=> 100000 => 1
[[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 110000 => 1
[[5,1],[1]]
=> [[5,4],[4]]
=> [5,4]
=> 1010000 => 2
[[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 11000 => 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [4,3]
=> 101000 => 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [5,3]
=> 1001000 => 2
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 111000 => 1
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [4,3,3]
=> 1011000 => 2
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [4,4,3]
=> 1101000 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000991
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> [1] => 1
[[2],[]]
=> []
=> []
=> [1] => 1
[[1,1],[]]
=> []
=> []
=> [1] => 1
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3],[]]
=> []
=> []
=> [1] => 1
[[2,1],[]]
=> []
=> []
=> [1] => 1
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[1,1,1],[]]
=> []
=> []
=> [1] => 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4],[]]
=> []
=> []
=> [1] => 1
[[3,1],[]]
=> []
=> []
=> [1] => 1
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[2,2],[]]
=> []
=> []
=> [1] => 1
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[2,1,1],[]]
=> []
=> []
=> [1] => 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3
[[1,1,1,1],[]]
=> []
=> []
=> [1] => 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4
[[5],[]]
=> []
=> []
=> [1] => 1
[[4,1],[]]
=> []
=> []
=> [1] => 1
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,2],[]]
=> []
=> []
=> [1] => 1
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[3,1,1],[]]
=> []
=> []
=> [1] => 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 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. St000292The number of ascents of a binary word. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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. St000702The number of weak deficiencies 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. 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. St000522The number of 1-protected nodes of a rooted tree. 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001489The maximum of the number of descents and the number of inverse descents. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St000646The number of big ascents of a permutation. 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 $eA$. St001712The number of natural descents of a standard Young tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001621The number of atoms of a lattice. 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. St001330The hat guessing number of a graph.
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