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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => 0
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001465
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => 0
[]
=> [] => [] => ? = 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000214
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => [2,4,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [2,4,3,1] => [3,4,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,3,4,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => [1,3,5,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,3,5,4,2] => [1,4,5,2,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => [2,3,5,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => [2,4,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [3,2,5,4,1] => [2,4,5,1,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,2,4,5,1] => [2,5,1,3,4] => 0
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [1,7,3,6,5,4,2] => [1,3,5,6,4,7,2] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,3,6,5,7,2] => [1,7,4,3,6,5,2] => [1,4,3,6,5,7,2] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,3,6,7,2] => [1,7,5,4,3,6,2] => [1,4,5,3,6,7,2] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,3,7,2] => [1,7,6,4,5,3,2] => [1,4,5,6,3,7,2] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,5,6,4,7,3] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [4,2,3,5,6,7,1] => [2,3,7,1,4,5,6] => ? = 0
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,3,6,1,7] => [6,2,5,4,3,1,7] => [2,4,5,3,6,1,7] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7] => [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [5,3,2,4,6,7,1] => [3,2,4,7,1,5,6] => ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,2,5,4,6,1,7] => [6,3,2,5,4,1,7] => [3,2,5,4,6,1,7] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [4,3,2,5,6,7,1] => [3,2,7,1,4,5,6] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,2,5,1,6,7] => [5,4,3,2,1,6,7] => [3,4,2,5,1,6,7] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [3,4,2,5,1,7,6] => [5,4,3,2,1,7,6] => [3,4,2,5,1,7,6] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [3,4,2,5,6,1,7] => [6,4,3,2,5,1,7] => [3,4,2,5,6,1,7] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,4,5,2,6,1,7] => [6,5,3,4,2,1,7] => [3,4,5,2,6,1,7] => ? = 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,3,5,2,6,1,7] => [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,5,4,8] => [1,2,3,5,7,6,4,8] => [1,2,3,6,7,4,5,8] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => [1,2,3,5,7,6,8,4] => [1,2,3,6,8,4,5,7] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,8,5,6,7,4,3] => [1,2,7,4,5,6,8,3] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,5,4,8] => [1,3,2,5,7,6,4,8] => [1,3,2,6,7,4,5,8] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,7,5,4] => [1,3,2,5,7,6,8,4] => [1,3,2,6,8,4,5,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,3,2,6,7,8] => [1,3,5,4,2,6,7,8] => [1,4,5,2,3,6,7,8] => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,3,2,7,6,8] => [1,3,5,4,2,7,6,8] => [1,4,5,2,3,7,6,8] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,5,3,2,8,7,6] => [1,3,5,4,2,7,8,6] => [1,4,5,2,3,8,6,7] => ? = 0
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,3,8,7,6,2] => [1,6,5,4,3,7,8,2] => [1,4,5,3,8,2,6,7] => ? = 0
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,7,3,8,2] => [1,8,7,4,5,6,3,2] => [1,4,5,6,7,3,8,2] => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,4,6,5,3,2,7,8] => [1,3,5,4,6,2,7,8] => [1,4,6,2,3,5,7,8] => ? = 0
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,7,5,3,2] => [1,3,5,4,7,6,8,2] => [1,4,6,8,2,3,5,7] => ? = 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,6,5,3,2,8] => [1,3,5,4,6,7,2,8] => [1,4,7,2,3,5,6,8] => ? = 0
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,7,6,5,3,2] => [1,3,5,4,6,7,8,2] => [1,4,8,2,3,5,6,7] => ? = 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,4,6,7,3,8,2] => [1,8,7,5,4,6,3,2] => [1,5,4,6,7,3,8,2] => ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,4,7,3,8,2] => [1,8,7,6,5,4,3,2] => [1,5,6,4,7,3,8,2] => ? = 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,1,4,7,6,5,8,3] => [2,1,8,4,6,7,5,3] => [2,1,4,7,5,6,8,3] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? = 0
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,8,7,6,5] => [3,2,1,4,6,7,8,5] => [2,3,1,4,8,5,6,7] => ? = 0
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,5,8,4] => [3,2,1,8,7,6,5,4] => [2,3,1,6,7,5,8,4] => ? = 0
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,8,7,6,5,1] => [5,2,3,4,6,7,8,1] => [2,3,4,8,1,5,6,7] => ? = 0
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,8,7,6,5,4,1] => [4,2,3,5,6,7,8,1] => [2,3,8,1,4,5,6,7] => ? = 0
[1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,4,3,1,7,8,6,5] => [3,2,4,1,6,8,7,5] => [2,4,1,3,7,8,5,6] => ? = 0
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [2,4,6,5,7,3,8,1] => [8,2,7,4,6,5,3,1] => [2,4,6,5,7,3,8,1] => ? = 1
[1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,5,4,7,8,6,3,1] => [3,2,6,5,4,8,7,1] => [2,5,4,7,8,1,3,6] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [3,2,4,1,5,6,7,8] => ? = 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,2,8,7,6,5,4,1] => [4,3,2,5,6,7,8,1] => [3,2,8,1,4,5,6,7] => ? = 1
[1,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,4,2,1,6,8,7,5] => [2,4,3,1,7,6,8,5] => [3,4,1,2,6,8,5,7] => ? = 0
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,4,2,1,8,7,6,5] => [2,4,3,1,6,7,8,5] => [3,4,1,2,8,5,6,7] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,4,2,5,1,6,7,8] => [5,4,3,2,1,6,7,8] => [3,4,2,5,1,6,7,8] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,2,5,1,7,8,6] => [5,4,3,2,1,8,7,6] => [3,4,2,5,1,7,8,6] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,4,2,5,1,8,7,6] => [5,4,3,2,1,7,8,6] => [3,4,2,5,1,8,6,7] => ? = 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000502
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => {{1,3},{2}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,7,6] => [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8] => [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,5,4,8] => [1,2,3,7,4,6,5,8] => {{1},{2},{3},{4,7},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => [1,2,3,8,4,6,5,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => [1,2,3,7,4,5,6,8] => {{1},{2},{3},{4,7},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,8,7,4,5,6,3] => {{1},{2},{3,8},{4,7},{5},{6}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,5,8,7,6] => [1,3,2,4,5,8,6,7] => {{1},{2,3},{4},{5},{6,8},{7}}
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,8,7,6] => [1,3,2,5,4,8,6,7] => {{1},{2,3},{4,5},{6,8},{7}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,5,4,7,8] => [1,3,2,6,4,5,7,8] => {{1},{2,3},{4,6},{5},{7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,5,4,8] => [1,3,2,7,4,6,5,8] => {{1},{2,3},{4,7},{5},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,7,5,4] => [1,3,2,8,4,6,5,7] => {{1},{2,3},{4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,6,5,4,8] => [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,7},{5},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,7,6,5,4] => [1,3,2,8,4,5,6,7] => {{1},{2,3},{4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7,8] => [1,4,2,3,5,6,7,8] => {{1},{2,4},{3},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,4,3,2,5,7,6,8] => [1,4,2,3,5,7,6,8] => {{1},{2,4},{3},{5},{6,7},{8}}
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,3,2,5,8,7,6] => [1,4,2,3,5,8,6,7] => {{1},{2,4},{3},{5},{6,8},{7}}
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => [1,4,2,3,7,5,6,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 0
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,3,2,6,7,8] => [1,5,2,4,3,6,7,8] => {{1},{2,5},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,3,2,7,6,8] => [1,5,2,4,3,7,6,8] => {{1},{2,5},{3},{4},{6,7},{8}}
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,5,3,2,8,7,6] => [1,5,2,4,3,8,6,7] => {{1},{2,5},{3},{4},{6,8},{7}}
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,3,8,7,6,2] => [1,8,5,4,3,2,6,7] => {{1},{2,8},{3,5},{4},{6},{7}}
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,7,3,8,2] => [1,8,7,4,5,6,3,2] => ?
=> ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,4,6,5,3,2,7,8] => [1,6,2,4,3,5,7,8] => {{1},{2,6},{3},{4},{5},{7},{8}}
=> ? = 0
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,7,5,3,2] => [1,8,2,4,3,6,5,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,6,5,3,2,8] => [1,7,2,4,3,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,7,6,5,3,2] => [1,8,2,4,3,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,5,4,3,2,6,7,8] => [1,5,2,3,4,6,7,8] => {{1},{2,5},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => [1,5,2,3,4,7,6,8] => {{1},{2,5},{3},{4},{6,7},{8}}
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,4,3,2,8,7,6] => [1,5,2,3,4,8,6,7] => {{1},{2,5},{3},{4},{6,8},{7}}
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,4,6,7,3,8,2] => [1,8,7,5,4,6,3,2] => ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,4,7,3,8,2] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8] => [1,6,2,3,4,5,7,8] => {{1},{2,6},{3},{4},{5},{7},{8}}
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,6,7,5,4,3,2,8] => [1,7,2,3,4,6,5,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,8,2,3,4,6,5,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,7,6,5,4,3,2,8] => [1,7,2,3,4,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0
Description
The number of successions of a set partitions.
This is the number of indices i such that i and i+1 belonging to the same block.
Matching statistic: St001061
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,5,6,4,7,3] => [2,1,7,6,5,4,3] => [2,1,5,6,7,3,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => [2,1,6,3,4,5,7] => [2,1,6,3,4,5,7] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => ? = 0
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [7,2,3,4,1,5,6] => [2,3,4,7,1,5,6] => ? = 0
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [7,2,3,1,4,5,6] => [2,3,7,1,4,5,6] => ? = 0
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,3,6,1,7] => [6,2,5,4,3,1,7] => [2,4,5,6,1,3,7] => ? = 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7] => [4,3,2,1,5,6,7] => [3,4,1,2,5,6,7] => ? = 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [7,3,2,4,1,5,6] => [3,4,1,7,2,5,6] => ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,2,5,4,6,1,7] => [6,3,2,5,4,1,7] => [3,5,1,6,2,4,7] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [7,3,2,1,4,5,6] => [3,7,1,2,4,5,6] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,2,5,1,6,7] => [5,4,3,2,1,6,7] => [3,4,5,1,2,6,7] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [3,4,2,5,1,7,6] => [5,4,3,2,1,7,6] => [3,4,5,1,2,7,6] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [3,4,2,5,6,1,7] => [6,4,3,2,5,1,7] => [3,4,5,1,6,2,7] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,4,5,2,6,1,7] => [6,5,3,4,2,1,7] => [3,4,5,6,1,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,3,5,2,6,1,7] => [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [7,6,5,4,3,2,1] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,7,6] => [1,2,3,5,4,8,6,7] => [1,2,3,5,4,8,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8] => [1,2,3,6,4,5,7,8] => [1,2,3,6,4,5,7,8] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,5,4,8] => [1,2,3,7,4,6,5,8] => [1,2,3,6,4,7,5,8] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => [1,2,3,8,4,6,5,7] => [1,2,3,6,4,8,5,7] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => [1,2,3,7,4,5,6,8] => [1,2,3,7,4,5,6,8] => ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [1,2,3,8,4,5,6,7] => [1,2,3,8,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,8,7,4,5,6,3] => [1,2,7,8,3,4,5,6] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => ? = 2
Description
The number of indices that are both descents and recoils of a permutation.
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