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Your data matches 105 different statistics following compositions of up to 3 maps.
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Matching statistic: St000486
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
[1,4,5,3,2] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000660
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> 0
[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]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[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]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001086
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => 0
[2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => 0
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 1
[1,4,5,3,2] => [1,5,4,3,2] => 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern 132, where the matched entries are all adjacent.
Matching statistic: St000023
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000647
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
Description
The number of big descents of a permutation.
For a permutation π, this is the number of indices i such that π(i)−π(i+1)>1.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000710
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation π is an index i such that i−π(i)>1.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000711
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation π is an index i such that π(i)−i>1.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000779
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
Description
The tier of a permutation.
This is the number of elements i such that [i+1,k,i] is an occurrence of the pattern [2,3,1]. For example, [3,5,6,1,2,4] has tier 2, with witnesses [3,5,2] (or [3,6,2]) and [5,6,4].
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000836
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
Description
The number of descents of distance 2 of a permutation.
This is, des2(π)=|{i:π(i)>π(i+2)}|.
Matching statistic: St000099
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 1 = 0 + 1
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 0 + 1
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 0 + 1
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000259The diameter of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001330The hat guessing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001597The Frobenius rank of a skew partition. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000934The 2-degree of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001820The size of the image of the pop stack sorting operator. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001875The number of simple modules with projective dimension at most 1. St000534The number of 2-rises of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001862The number of crossings of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000758The length of the longest staircase fitting into an integer composition. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph.
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