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Your data matches 182 different statistics following compositions of up to 3 maps.
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Matching statistic: St000542
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => 3
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => 2
[2,3,1] => [2,1,3] => 2
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 1
[1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [4,3,1,2] => 3
[1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [4,2,1,3] => 3
[1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [4,1,2,3] => 2
[2,1,3,4] => [3,4,2,1] => 3
[2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,4,1] => 3
[2,3,4,1] => [3,2,1,4] => 3
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [3,1,2,4] => 2
[3,1,2,4] => [2,4,3,1] => 2
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [2,3,4,1] => 2
[3,2,4,1] => [2,3,1,4] => 2
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 2
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [5,4,3,1,2] => 4
[1,2,4,3,5] => [5,4,2,3,1] => 4
[1,2,4,5,3] => [5,4,2,1,3] => 4
[1,2,5,3,4] => [5,4,1,3,2] => 3
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [5,3,4,1,2] => 3
[1,3,4,2,5] => [5,3,2,4,1] => 4
[1,3,4,5,2] => [5,3,2,1,4] => 4
[1,3,5,2,4] => [5,3,1,4,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 3
[1,4,2,3,5] => [5,2,4,3,1] => 3
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [5,2,3,1,4] => 3
[1,4,5,2,3] => [5,2,1,4,3] => 3
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St001068
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000010
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,1]
=> 2
[2,1] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [3,2,1] => [2,1]
=> 2
[3,1,2] => [2,3,1] => [3]
=> 1
[3,2,1] => [3,1,2] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3
[1,4,2,3] => [1,3,4,2] => [3,1]
=> 2
[1,4,3,2] => [1,4,2,3] => [3,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[2,4,1,3] => [3,2,4,1] => [3,1]
=> 2
[2,4,3,1] => [4,2,1,3] => [3,1]
=> 2
[3,1,2,4] => [2,3,1,4] => [3,1]
=> 2
[3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[3,2,1,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[3,4,1,2] => [2,4,3,1] => [3,1]
=> 2
[3,4,2,1] => [4,1,3,2] => [3,1]
=> 2
[4,1,2,3] => [2,3,4,1] => [4]
=> 1
[4,1,3,2] => [3,4,2,1] => [4]
=> 1
[4,2,1,3] => [3,1,4,2] => [4]
=> 1
[4,2,3,1] => [4,3,1,2] => [4]
=> 1
[4,3,1,2] => [2,4,1,3] => [4]
=> 1
[4,3,2,1] => [4,1,2,3] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 4
[1,2,5,3,4] => [1,2,4,5,3] => [3,1,1]
=> 3
[1,2,5,4,3] => [1,2,5,3,4] => [3,1,1]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 4
[1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 3
[1,3,5,4,2] => [1,5,3,2,4] => [3,1,1]
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,1]
=> 3
[1,4,2,5,3] => [1,4,5,2,3] => [2,2,1]
=> 3
[1,4,3,2,5] => [1,4,2,3,5] => [3,1,1]
=> 3
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> 3
[1,4,5,2,3] => [1,3,5,4,2] => [3,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000011
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000105
(load all 62 compositions to match this statistic)
(load all 62 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima 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%
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] => [1,0]
=> {{1}}
=> 1
[1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
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: St000167
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[]]
=> 1
[1,2] => [1,0,1,0]
=> [[],[]]
=> 2
[2,1] => [1,1,0,0]
=> [[[]]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 3
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000318
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ 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
[1,2] => [1,0,1,0]
=> [1]
=> 2
[2,1] => [1,1,0,0]
=> []
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> []
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> []
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000470
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 2
[2,1] => [1,1,0,0]
=> [1,2] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000912
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ([],1)
=> 1
[1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> ([],2)
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> ([],3)
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> ([],3)
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
Description
The number of maximal antichains in a poset.
The following 172 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St000340The number of non-final maximal constant sub-paths of length greater than one. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000007The number of saliances of the permutation. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000528The height of a poset. St000676The number of odd rises of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000925The number of topologically connected components of a set partition. St001029The size of the core of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000141The maximum drop size of a permutation. St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000439The position of the first down step of a Dyck path. St000536The pathwidth of a graph. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000354The number of recoils of a permutation. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000159The number of distinct parts of the integer partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000237The number of small exceedances. St000989The number of final rises of a permutation. St000546The number of global descents of a permutation. St000054The first entry of the permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000991The number of right-to-left minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000164The number of short pairs. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000325The width of the tree associated to a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001180Number of indecomposable injective modules with projective dimension at most 1. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000286The number of connected components of the complement of a graph. St000308The height of the tree associated to a permutation. St000389The number of runs of ones of odd length in a binary word. St000740The last entry of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000203The number of external nodes of a binary tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001330The hat guessing number of a graph. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000236The number of cyclical small weak excedances. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001152The number of pairs with even minimum in a perfect matching. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St000710The number of big deficiencies of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. 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 $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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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