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Your data matches 105 different statistics following compositions of up to 3 maps.
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Matching statistic: St001604
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [3,2,1]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000406
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of occurrences of the pattern 3241 in a permutation.
Matching statistic: St000622
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000622: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000622: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of occurrences of the patterns 2143 or 4231 in a permutation.
It is a necessary and sufficient condition that a permutation $\pi$ avoids these two patterns for the Schubert variety associated to $\pi$ to be smooth [1].
Matching statistic: St000623
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of occurrences of the pattern 52341 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to have a complete parabolic bundle structure [1].
Matching statistic: St000666
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of right tethers of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right tether is a large ascent between two consecutive rafts of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000750
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000750: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000750: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of occurrences of the pattern 4213 in a permutation.
Matching statistic: St000800
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000962
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000962: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000962: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The 3-shifted major index of a permutation.
This is given by the sum of all indices $i$ such that $\pi(i)-\pi(i+1) > 3$.
Summing with [[St000494]] yields Rawlings' Mahonian statistic, see [1, p. 50].
Matching statistic: St001130
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The number of two successive successions in a permutation.
Matching statistic: St001381
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001381: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001381: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => [1,5,6,4,3,2,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [6,5,1,4,3,2,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [1,5,2,6,4,3,7] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,6,2,5,4,3,7] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => [7,6,5,4,1,3,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [6,5,4,1,7,3,2] => [1,5,4,3,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [7,5,4,1,6,3,2] => [1,6,4,3,5,2,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [7,3,6,5,1,4,2] => [6,2,5,4,3,7,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [7,6,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => [1,6,7,5,4,3,2,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,7,6,5,4,3,2] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,8,7,5,4,3,2] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,8,7,6,4,3,2] => ? => ? = 1
Description
The fertility of a permutation.
This is the size of the preimage of a permutation $\pi$ under the stack-sorting map $s$ defined by Julian West.
For a partial permutation $\pi = \pi_1 \cdots \pi_n$ define $s(\pi)$ recursively as follows:
* If the $n\leq 1$ set $s(\pi) = \pi$.
* Otherwise, let $m$ be the largest entry in $\pi$, write $\pi=LmR$ and define $s(\pi) = s(L)s(R)m$.
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000210Minimum over maximum difference of elements in cycles. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000487The length of the shortest cycle of a permutation. St001545The second Elser number of a connected graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000914The sum of the values of the Möbius function of a poset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001850The number of Hecke atoms of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St001810The number of fixed points of a permutation smaller than its largest moved point. St000694The number of affine bounded permutations that project to a given permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St000401The size of the symmetry class of a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000717The number of ordinal summands of a poset. St001260The permanent of an alternating sign matrix. St000124The cardinality of the preimage of the Simion-Schmidt map. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000264The girth of a graph, which is not a tree. St000629The defect of a binary word. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000297The number of leading ones in a binary word. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001820The size of the image of the pop stack sorting operator. St000298The order dimension or Dushnik-Miller dimension of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000862The number of parts of the shifted shape of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001928The number of non-overlapping descents in a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000322The skewness of a graph. St001665The number of pure excedances of a permutation. St000042The number of crossings of a perfect matching. St000296The length of the symmetric border of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St000402Half the size of the symmetry class of a permutation. St000664The number of right ropes of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001434The number of negative sum pairs of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000353The number of inner valleys of a permutation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000908The length of the shortest maximal antichain in a poset. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001403The number of vertical separators in a permutation.
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