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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000053
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001499
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
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%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001068
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000024
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000340
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000105
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2 = 1 + 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3 = 2 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4 = 3 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4 = 3 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2 = 1 + 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 3 = 2 + 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> 3 = 2 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000925
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2 = 1 + 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3 = 2 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4 = 3 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4 = 3 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2 = 1 + 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 3 = 2 + 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> 3 = 2 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St001007
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001007: 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
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001036
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: 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
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000292
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 10 => 01 => 1
[[2],[]]
=> [2]
=> 100 => 001 => 1
[[1,1],[]]
=> [1,1]
=> 110 => 101 => 1
[[2,1],[1]]
=> [2,1]
=> 1010 => 0101 => 2
[[3],[]]
=> [3]
=> 1000 => 0001 => 1
[[2,1],[]]
=> [2,1]
=> 1010 => 0101 => 2
[[3,1],[1]]
=> [3,1]
=> 10010 => 00101 => 2
[[2,2],[1]]
=> [2,2]
=> 1100 => 1001 => 1
[[3,2],[2]]
=> [3,2]
=> 10100 => 01001 => 2
[[1,1,1],[]]
=> [1,1,1]
=> 1110 => 1101 => 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> 11010 => 10101 => 2
[[2,1,1],[1]]
=> [2,1,1]
=> 10110 => 01101 => 2
[[3,2,1],[2,1]]
=> [3,2,1]
=> 101010 => 010101 => 3
[[4],[]]
=> [4]
=> 10000 => 00001 => 1
[[3,1],[]]
=> [3,1]
=> 10010 => 00101 => 2
[[4,1],[1]]
=> [4,1]
=> 100010 => 000101 => 2
[[2,2],[]]
=> [2,2]
=> 1100 => 1001 => 1
[[3,2],[1]]
=> [3,2]
=> 10100 => 01001 => 2
[[4,2],[2]]
=> [4,2]
=> 100100 => 001001 => 2
[[2,1,1],[]]
=> [2,1,1]
=> 10110 => 01101 => 2
[[3,2,1],[1,1]]
=> [3,2,1]
=> 101010 => 010101 => 3
[[3,1,1],[1]]
=> [3,1,1]
=> 100110 => 001101 => 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> 1001010 => 0010101 => 3
[[3,3],[2]]
=> [3,3]
=> 11000 => 10001 => 1
[[4,3],[3]]
=> [4,3]
=> 101000 => 010001 => 2
[[2,2,1],[1]]
=> [2,2,1]
=> 11010 => 10101 => 2
[[3,3,1],[2,1]]
=> [3,3,1]
=> 110010 => 100101 => 2
[[3,2,1],[2]]
=> [3,2,1]
=> 101010 => 010101 => 3
[[4,3,1],[3,1]]
=> [4,3,1]
=> 1010010 => 0100101 => 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> 11100 => 11001 => 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> 110100 => 101001 => 2
[[3,2,2],[2,1]]
=> [3,2,2]
=> 101100 => 011001 => 2
[[4,3,2],[3,2]]
=> [4,3,2]
=> 1010100 => 0101001 => 3
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 11101 => 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> 111010 => 110101 => 2
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> 110110 => 101101 => 2
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> 1101010 => 1010101 => 3
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> 101110 => 011101 => 2
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> 1011010 => 0110101 => 3
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> 1010110 => 0101101 => 3
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> 10101010 => 01010101 => 4
[[5],[]]
=> [5]
=> 100000 => 000001 => 1
[[4,1],[]]
=> [4,1]
=> 100010 => 000101 => 2
[[5,1],[1]]
=> [5,1]
=> 1000010 => 0000101 => 2
[[3,2],[]]
=> [3,2]
=> 10100 => 01001 => 2
[[4,2],[1]]
=> [4,2]
=> 100100 => 001001 => 2
[[5,2],[2]]
=> [5,2]
=> 1000100 => 0001001 => 2
[[3,1,1],[]]
=> [3,1,1]
=> 100110 => 001101 => 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> 1001010 => 0010101 => 3
[[4,1,1],[1]]
=> [4,1,1]
=> 1000110 => 0001101 => 2
[[6,4,3,2,1],[4,3,2,1]]
=> [6,4,3,2,1]
=> 10010101010 => 00101010101 => ? = 5
[[6,5,2,1],[5,2,1]]
=> [6,5,2,1]
=> 1010001010 => 0100010101 => ? = 4
[[6,5,3,1],[5,3,1]]
=> [6,5,3,1]
=> 1010010010 => 0100100101 => ? = 4
[[6,5,3,2],[5,3,2]]
=> [6,5,3,2]
=> 1010010100 => 0100101001 => ? = 4
[[6,5,4,1],[5,4,1]]
=> [6,5,4,1]
=> 1010100010 => 0101000101 => ? = 4
[[6,5,4,2],[5,4,2]]
=> [6,5,4,2]
=> 1010100100 => 0101001001 => ? = 4
[[6,5,4,3],[5,4,3]]
=> [6,5,4,3]
=> 1010101000 => 0101010001 => ? = 4
[[4,4,4,3,2,1],[3,3,3,2,1]]
=> [4,4,4,3,2,1]
=> 1110101010 => 1101010101 => ? = 4
[[4,4,3,3,2,1],[3,3,2,2,1]]
=> [4,4,3,3,2,1]
=> 1101101010 => ? => ? = 4
[[4,4,3,2,2,1],[3,3,2,1,1]]
=> [4,4,3,2,2,1]
=> 1101011010 => ? => ? = 4
[[4,4,3,2,1,1],[3,3,2,1]]
=> [4,4,3,2,1,1]
=> 1101010110 => ? => ? = 4
[[4,3,3,3,2,1],[3,2,2,2,1]]
=> [4,3,3,3,2,1]
=> 1011101010 => ? => ? = 4
[[4,3,3,2,2,1],[3,2,2,1,1]]
=> [4,3,3,2,2,1]
=> 1011011010 => ? => ? = 4
[[4,3,3,2,1,1],[3,2,2,1]]
=> [4,3,3,2,1,1]
=> 1011010110 => ? => ? = 4
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [5,4,4,3,2,1]
=> 10110101010 => 01101010101 => ? = 5
[[4,3,2,2,2,1],[3,2,1,1,1]]
=> [4,3,2,2,2,1]
=> 1010111010 => ? => ? = 4
[[4,3,2,2,1,1],[3,2,1,1]]
=> [4,3,2,2,1,1]
=> 1010110110 => ? => ? = 4
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [5,4,3,3,2,1]
=> 10101101010 => 01011010101 => ? = 5
[[4,3,2,1,1,1],[3,2,1]]
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 4
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [5,4,3,2,2,1]
=> 10101011010 => 01010110101 => ? = 5
[[5,4,3,2,1,1],[4,3,2,1]]
=> [5,4,3,2,1,1]
=> 10101010110 => 01010101101 => ? = 5
[[6,2,2,1],[2,1,1]]
=> [6,2,2,1]
=> 1000011010 => ? => ? = 3
[[6,3,1,1],[3,1]]
=> [6,3,1,1]
=> 1000100110 => ? => ? = 3
[[6,4,3,2,1],[3,3,2,1]]
=> [6,4,3,2,1]
=> 10010101010 => 00101010101 => ? = 5
[[6,3,3,2,1],[3,2,2,1]]
=> [6,3,3,2,1]
=> 10001101010 => ? => ? = 4
[[6,3,2,2,1],[3,2,1,1]]
=> [6,3,2,2,1]
=> 10001011010 => ? => ? = 4
[[6,3,2,1,1],[3,2,1]]
=> [6,3,2,1,1]
=> 10001010110 => ? => ? = 4
[[6,5,2,1],[4,2,1]]
=> [6,5,2,1]
=> 1010001010 => 0100010101 => ? = 4
[[6,5,3,1],[4,3,1]]
=> [6,5,3,1]
=> 1010010010 => 0100100101 => ? = 4
[[6,5,3,2],[4,3,2]]
=> [6,5,3,2]
=> 1010010100 => 0100101001 => ? = 4
[[6,4,3,2,1],[4,2,2,1]]
=> [6,4,3,2,1]
=> 10010101010 => 00101010101 => ? = 5
[[6,4,2,2,1],[4,2,1,1]]
=> [6,4,2,2,1]
=> 10010011010 => ? => ? = 4
[[6,4,2,1,1],[4,2,1]]
=> [6,4,2,1,1]
=> 10010010110 => ? => ? = 4
[[6,5,4,1],[4,4,1]]
=> [6,5,4,1]
=> 1010100010 => 0101000101 => ? = 4
[[6,4,4,1],[4,3,1]]
=> [6,4,4,1]
=> 1001100010 => 0011000101 => ? = 3
[[6,5,4,2],[4,4,2]]
=> [6,5,4,2]
=> 1010100100 => 0101001001 => ? = 4
[[6,4,4,2],[4,3,2]]
=> [6,4,4,2]
=> 1001100100 => 0011001001 => ? = 3
[[6,4,4,2,1],[4,3,2,1]]
=> [6,4,4,2,1]
=> 10011001010 => ? => ? = 4
[[6,4,3,2,1],[4,3,1,1]]
=> [6,4,3,2,1]
=> 10010101010 => 00101010101 => ? = 5
[[6,4,3,1,1],[4,3,1]]
=> [6,4,3,1,1]
=> 10010100110 => ? => ? = 4
[[6,5,4,3],[4,4,3]]
=> [6,5,4,3]
=> 1010101000 => 0101010001 => ? = 4
[[6,4,4,3],[4,3,3]]
=> [6,4,4,3]
=> 1001101000 => 0011010001 => ? = 3
[[5,3,3,3,1],[3,2,2,1]]
=> [5,3,3,3,1]
=> 1001110010 => ? => ? = 3
[[6,4,4,3,1],[4,3,3,1]]
=> [6,4,4,3,1]
=> 10011010010 => ? => ? = 4
[[6,4,3,3,1],[4,3,2,1]]
=> [6,4,3,3,1]
=> 10010110010 => ? => ? = 4
[[6,4,3,2,1],[4,3,2]]
=> [6,4,3,2,1]
=> 10010101010 => 00101010101 => ? = 5
[[5,3,3,3,2],[3,2,2,2]]
=> [5,3,3,3,2]
=> 1001110100 => 0011101001 => ? = 3
[[5,3,3,2,2],[3,2,2,1]]
=> [5,3,3,2,2]
=> 1001101100 => 0011011001 => ? = 3
[[6,4,4,3,2],[4,3,3,2]]
=> [6,4,4,3,2]
=> 10011010100 => ? => ? = 4
[[5,3,2,2,2],[3,2,1,1]]
=> [5,3,2,2,2]
=> 1001011100 => ? => ? = 3
Description
The number of ascents of a binary word.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000159The number of distinct parts of the integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000015The number of peaks of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St000245The number of ascents of a permutation. St000069The number of maximal elements of a poset. St000167The number of leaves of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000068The number of minimal elements in a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000386The number of factors DDU in a Dyck path. St000021The number of descents 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. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001083The number of boxed occurrences of 132 in a permutation. 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. St000764The number of strong records in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of 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. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St001712The number of natural descents of a standard Young tableau. 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 $eA$. 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.
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