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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001086
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] => [1] => 0
[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
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: St000660
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 0
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[7,5,6,4,3,8,2,1] => [1,2,8,3,4,6,5,7] => [1,2,4,5,7,6,8,3] => ?
=> ? = 0
[8,5,4,3,2,6,7,1] => [1,7,6,2,3,4,5,8] => [1,4,5,6,7,3,2,8] => ?
=> ? = 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 1
[8,7,6,5,2,1,3,4] => [4,3,1,2,5,6,7,8] => [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,7,6,3,4,1,2,5] => [5,2,1,4,3,6,7,8] => [3,2,5,4,1,6,7,8] => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[7,8,4,2,3,1,5,6] => [6,5,1,3,2,4,8,7] => [3,5,4,6,2,1,8,7] => ?
=> ? = 1
[8,5,6,4,2,1,3,7] => [7,3,1,2,4,6,5,8] => [3,4,2,5,7,6,1,8] => [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 1
[8,5,6,3,2,1,4,7] => [7,4,1,2,3,6,5,8] => [3,4,5,2,7,6,1,8] => [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 1
[7,6,4,3,5,2,1,8] => [8,1,2,5,3,4,6,7] => [2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
[5,6,7,3,2,4,1,8] => [8,1,4,2,3,7,6,5] => [2,4,5,3,8,7,6,1] => [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[7,5,4,3,2,6,1,8] => [8,1,6,2,3,4,5,7] => [2,4,5,6,7,3,8,1] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 0
[7,5,3,4,2,6,1,8] => [8,1,6,2,4,3,5,7] => [2,4,6,5,7,3,8,1] => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[7,4,5,2,3,6,1,8] => [8,1,6,3,2,5,4,7] => [2,5,4,7,6,3,8,1] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[6,5,3,2,4,7,1,8] => [8,1,7,4,2,3,5,6] => [2,5,6,4,7,8,3,1] => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1
[7,5,3,1,2,4,6,8] => [8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[5,6,4,3,1,2,7,8] => [8,7,2,1,3,4,6,5] => [4,3,5,6,8,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
[5,4,6,2,1,3,7,8] => [8,7,3,1,2,6,4,5] => [4,5,3,7,8,6,2,1] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 1
[6,3,4,2,1,5,7,8] => [8,7,5,1,2,4,3,6] => [4,5,7,6,3,8,2,1] => ?
=> ? = 1
[5,4,3,2,1,6,7,8] => [8,7,6,1,2,3,4,5] => [4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,5,6,4,7,8,3,2] => [2,3,8,7,4,6,5,1] => [8,1,2,5,7,6,4,3] => ?
=> ? = 1
[1,3,7,8,6,5,4,2] => [2,4,5,6,8,7,3,1] => [8,1,7,2,3,4,6,5] => ?
=> ? = 1
[1,4,6,5,3,8,7,2] => [2,7,8,3,5,6,4,1] => [8,1,4,7,5,6,2,3] => ?
=> ? = 1
[1,4,5,3,6,8,7,2] => [2,7,8,6,3,5,4,1] => [8,1,5,7,6,4,2,3] => ?
=> ? = 1
[1,6,5,7,4,3,8,2] => [2,8,3,4,7,5,6,1] => [8,1,3,4,6,7,5,2] => ?
=> ? = 1
[1,5,6,7,4,3,8,2] => [2,8,3,4,7,6,5,1] => [8,1,3,4,7,6,5,2] => ?
=> ? = 1
[2,1,5,7,6,8,4,3] => [3,4,8,6,7,5,1,2] => [7,8,1,2,6,4,5,3] => ?
=> ? = 1
[2,1,5,4,7,8,6,3] => [3,6,8,7,4,5,1,2] => [7,8,1,5,6,2,4,3] => ?
=> ? = 1
[2,1,5,6,4,8,7,3] => [3,7,8,4,6,5,1,2] => [7,8,1,4,6,5,2,3] => ?
=> ? = 1
[2,1,4,5,6,8,7,3] => [3,7,8,6,5,4,1,2] => [7,8,1,6,5,4,2,3] => ?
=> ? = 1
[2,1,6,7,5,4,8,3] => [3,8,4,5,7,6,1,2] => [7,8,1,3,4,6,5,2] => ?
=> ? = 1
[2,1,5,6,7,4,8,3] => [3,8,4,7,6,5,1,2] => [7,8,1,3,6,5,4,2] => ?
=> ? = 1
[1,2,7,6,8,5,4,3] => [3,4,5,8,6,7,2,1] => [8,7,1,2,3,5,6,4] => ?
=> ? = 1
[1,2,6,5,8,7,4,3] => [3,4,7,8,5,6,2,1] => [8,7,1,2,5,6,3,4] => ?
=> ? = 1
[1,2,5,7,6,8,4,3] => [3,4,8,6,7,5,2,1] => [8,7,1,2,6,4,5,3] => ?
=> ? = 1
[1,2,6,5,7,8,4,3] => [3,4,8,7,5,6,2,1] => [8,7,1,2,5,6,4,3] => ?
=> ? = 1
[1,2,5,4,6,8,7,3] => [3,7,8,6,4,5,2,1] => [8,7,1,5,6,4,2,3] => ?
=> ? = 1
[1,2,4,5,6,8,7,3] => [3,7,8,6,5,4,2,1] => [8,7,1,6,5,4,2,3] => ?
=> ? = 1
[1,2,5,7,6,4,8,3] => [3,8,4,6,7,5,2,1] => [8,7,1,3,6,4,5,2] => ?
=> ? = 1
[1,2,5,4,7,6,8,3] => [3,8,6,7,4,5,2,1] => [8,7,1,5,6,3,4,2] => ?
=> ? = 1
[1,3,2,6,7,8,5,4] => [4,5,8,7,6,2,3,1] => [8,6,7,1,2,5,4,3] => ?
=> ? = 1
[1,3,2,5,6,8,7,4] => [4,7,8,6,5,2,3,1] => [8,6,7,1,5,4,2,3] => ?
=> ? = 1
[2,1,3,7,8,6,5,4] => [4,5,6,8,7,3,1,2] => [7,8,6,1,2,3,5,4] => ?
=> ? = 1
[2,1,3,6,8,7,5,4] => [4,5,7,8,6,3,1,2] => [7,8,6,1,2,5,3,4] => ?
=> ? = 1
[2,1,3,6,7,8,5,4] => [4,5,8,7,6,3,1,2] => [7,8,6,1,2,5,4,3] => ?
=> ? = 1
[1,4,3,2,7,6,8,5] => [5,8,6,7,2,3,4,1] => [8,5,6,7,1,3,4,2] => ?
=> ? = 1
[1,3,4,2,7,6,8,5] => [5,8,6,7,2,4,3,1] => [8,5,7,6,1,3,4,2] => ?
=> ? = 1
[3,2,1,4,7,6,8,5] => [5,8,6,7,4,1,2,3] => [6,7,8,5,1,3,4,2] => ?
=> ? = 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: St000647
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => 0
[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
[7,8,5,6,4,3,2,1] => [7,8,5,6,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ? = 0
[6,5,7,8,4,3,2,1] => [6,5,8,7,4,3,2,1] => [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,3,4,2,5,6,7,8] => ? = 1
[7,8,6,4,5,3,2,1] => [7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ? = 0
[8,6,4,5,7,3,2,1] => [8,6,4,7,5,3,2,1] => [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ? = 1
[8,5,4,6,7,3,2,1] => [8,5,4,7,6,3,2,1] => [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ? = 1
[7,8,6,5,3,4,2,1] => [7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [2,1,3,4,6,5,7,8] => ? = 0
[8,6,7,5,3,4,2,1] => [8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ? = 0
[8,7,4,3,5,6,2,1] => [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,2,3,5,6,4,7,8] => ? = 1
[7,5,6,4,3,8,2,1] => [7,5,8,4,3,6,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ? = 0
[7,8,6,5,4,2,3,1] => [7,8,6,5,4,2,3,1] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [2,1,3,4,5,7,6,8] => ? = 0
[8,7,5,6,4,2,3,1] => [8,7,5,6,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,2,4,3,5,7,6,8] => ? = 0
[8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,2,4,3,5,7,6,8] => ? = 0
[8,7,6,3,2,4,5,1] => [8,7,6,3,2,5,4,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,2,3,4,6,7,5,8] => ? = 1
[8,6,5,4,2,3,7,1] => [8,6,5,4,2,7,3,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,2,3,4,6,7,5,8] => ? = 1
[8,5,4,3,2,6,7,1] => [8,5,4,3,2,7,6,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,2,3,4,6,7,5,8] => ? = 1
[8,4,5,2,3,6,7,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,3,2,5,6,7,4,8] => ? = 1
[8,3,2,4,5,6,7,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,2,4,5,6,7,3,8] => ? = 1
[6,5,7,4,2,3,8,1] => [6,5,8,4,2,7,3,1] => [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> [1,3,2,4,6,7,5,8] => ? = 1
[6,2,3,4,5,7,8,1] => [6,2,8,7,5,4,3,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,3,4,5,6,7,2,8] => ? = 1
[8,6,7,5,4,3,1,2] => [8,6,7,5,4,3,1,2] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,3,2,4,5,6,8,7] => ? = 0
[8,7,5,6,4,3,1,2] => [8,7,5,6,4,3,1,2] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => ? = 0
[8,7,6,4,5,3,1,2] => [8,7,6,4,5,3,1,2] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ? = 0
[8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ? = 0
[8,7,5,6,3,4,1,2] => [8,7,5,6,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,2,4,3,6,5,8,7] => ? = 0
[6,5,7,8,3,4,1,2] => [6,5,8,7,3,4,1,2] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,3,4,2,6,5,8,7] => ? = 1
[7,8,4,3,5,6,1,2] => [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [2,1,3,5,6,4,8,7] => ? = 1
[8,5,4,6,3,7,1,2] => [8,5,4,7,3,6,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,2,4,3,6,5,8,7] => ? = 0
[5,4,3,6,7,8,1,2] => [5,4,3,8,7,6,1,2] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,2,4,5,6,3,8,7] => ? = 1
[7,6,8,5,4,2,1,3] => [7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,3,2,4,5,6,8,7] => ? = 0
[8,4,5,6,7,1,2,3] => [8,4,7,6,5,1,3,2] => [[[.,.],[[[.,.],.],.]],[[.,.],.]]
=> [1,3,4,5,2,7,8,6] => ? = 2
[7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => ? = 0
[7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,3,4,2,6,5,8,7] => ? = 1
[6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [2,1,4,3,5,7,8,6] => ? = 1
[8,7,6,5,2,1,3,4] => [8,7,6,5,2,1,4,3] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => ? = 1
[7,8,5,6,2,1,3,4] => [7,8,5,6,2,1,4,3] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [2,1,4,3,5,7,8,6] => ? = 1
[6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,3,4,2,5,7,8,6] => ? = 2
[8,7,6,3,4,1,2,5] => [8,7,6,3,5,1,4,2] => [[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,2,3,5,4,7,8,6] => ? = 1
[7,8,4,2,3,1,5,6] => [7,8,4,2,6,1,5,3] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [2,1,3,5,4,7,8,6] => ? = 1
[7,8,3,2,1,4,5,6] => [7,8,3,2,1,6,5,4] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [2,1,3,4,6,7,8,5] => ? = 1
[8,5,6,4,2,1,3,7] => [8,5,7,4,2,1,6,3] => [[[[[.,.],[.,.]],.],.],[[.,.],.]]
=> [1,3,2,4,5,7,8,6] => ? = 1
[8,5,6,3,2,1,4,7] => [8,5,7,3,2,1,6,4] => [[[[[.,.],[.,.]],.],.],[[.,.],.]]
=> [1,3,2,4,5,7,8,6] => ? = 1
[7,6,4,3,5,2,1,8] => [7,6,4,3,8,2,1,5] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ? = 0
[5,6,7,3,2,4,1,8] => [5,8,7,3,2,6,1,4] => [[[[.,[[.,.],.]],.],[.,.]],[.,.]]
=> [2,3,1,4,6,5,8,7] => ? = 1
[7,5,4,3,2,6,1,8] => [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ? = 0
[7,5,3,4,2,6,1,8] => [7,5,3,8,2,6,1,4] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,2,4,3,6,5,8,7] => ? = 0
[7,4,5,2,3,6,1,8] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,3,2,5,6,4,8,7] => ? = 1
[6,5,3,2,4,7,1,8] => [6,5,3,2,8,7,1,4] => [[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> [1,2,3,5,6,4,8,7] => ? = 1
[7,6,5,3,2,1,4,8] => [7,6,5,3,2,1,8,4] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => ? = 1
[7,1,2,3,4,5,6,8] => [7,1,8,6,5,4,3,2] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => ? = 1
[6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => ? = 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
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: St000486
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => ? = 0
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
Description
The number of cycles of length at least 3 of a permutation.
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: 11% ●values known / values provided: 11%●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: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => ? = 0
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
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
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => ? = 0
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 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
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => ? = 0
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
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: 10% ●values known / values provided: 10%●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: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => 0
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
[1,2,5,3,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
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: 10% ●values known / values provided: 10%●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: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1] => 1 = 0 + 1
[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
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000307
Mp00069: Permutations —complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,2,4,3] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,3,2,4] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,3,4,2] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,4,3,2] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,1,4,3] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,3,1,4] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[2,3,4,1] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[2,4,3,1] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,4,1] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,2,3,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,2,3,4] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,5,3,2,4] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,3,4,2] => [5,1,3,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,5,4,2,3] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[1,5,4,3,2] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,3,4,5] => [4,5,3,2,1] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[2,1,3,5,4] => [4,5,3,1,2] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[2,1,4,3,5] => [4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,4,5,3] => [4,5,2,1,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[2,1,5,3,4] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,5,4,3] => [4,5,1,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[2,3,1,4,5] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,3,1,5,4] => [4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,3,4,1,5] => [4,3,2,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 + 1
[2,3,4,5,1] => [4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,3,5,1,4] => [4,3,1,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 + 1
[2,3,5,4,1] => [4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,1,3,5] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,3,1,5] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 + 1
[2,4,3,5,1] => [4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[4,3,2,1,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,2,5,1] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,5,1,2] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,5,2,1] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,2,1,4] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,2,4,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,1,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,3,2,6,1] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,3,6,1,2] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,3,6,2,1] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,6,2,1,3] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,6,2,3,1] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,6,3,1,2] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,4,6,3,2,1] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,2,1,4] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,2,4,1] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
The following 1 statistic also match your data. Click on any of them to see the details.
St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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