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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000807
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Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[2] => [2] => 0
[1,1,1] => [1,1,1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[3] => [3] => 0
[1,1,1,1] => [1,1,1,1] => 0
[1,1,2] => [1,2,1] => 0
[1,2,1] => [2,1,1] => 0
[1,3] => [3,1] => 0
[2,1,1] => [1,1,2] => 0
[2,2] => [2,2] => 0
[3,1] => [1,3] => 0
[4] => [4] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 0
[1,1,1,2] => [1,1,2,1] => 0
[1,1,2,1] => [1,2,1,1] => 0
[1,1,3] => [1,3,1] => 0
[1,2,1,1] => [2,1,1,1] => 0
[1,2,2] => [2,2,1] => 0
[1,3,1] => [3,1,1] => 0
[1,4] => [4,1] => 0
[2,1,1,1] => [1,1,1,2] => 0
[2,1,2] => [1,2,2] => 0
[2,2,1] => [2,1,2] => 1
[2,3] => [3,2] => 0
[3,1,1] => [1,1,3] => 0
[3,2] => [2,3] => 0
[4,1] => [1,4] => 0
[5] => [5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[1,1,1,1,2] => [1,1,1,2,1] => 0
[1,1,1,2,1] => [1,1,2,1,1] => 0
[1,1,1,3] => [1,1,3,1] => 0
[1,1,2,1,1] => [1,2,1,1,1] => 0
[1,1,2,2] => [1,2,2,1] => 0
[1,1,3,1] => [1,3,1,1] => 0
[1,1,4] => [1,4,1] => 0
[1,2,1,1,1] => [2,1,1,1,1] => 0
[1,2,1,2] => [2,1,2,1] => 1
[1,2,2,1] => [2,2,1,1] => 0
[1,2,3] => [2,3,1] => 0
[1,3,1,1] => [3,1,1,1] => 0
[1,3,2] => [3,2,1] => 0
[1,4,1] => [4,1,1] => 0
[1,5] => [5,1] => 0
[2,1,1,1,1] => [1,1,1,1,2] => 0
[2,1,1,2] => [1,1,2,2] => 0
[2,1,2,1] => [1,2,1,2] => 1
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000405
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(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [1,0]
=> [2,1] => 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? = 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [8,3,7,5,6,1,2,4] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,8,7,5,6,1,3,4] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,6,5,1,2,3,4] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,8,6,7,1,5] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [8,3,4,7,6,1,2,5] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,4,7,6,1,3,5] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,8,4,6,1,2,3,5] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,3,8,7,6,1,4,5] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,8,6,1,2,4,5] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [2,3,8,5,7,1,4,6] => ? = 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [7,3,8,5,1,2,4,6] => ? = 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [2,7,8,5,1,3,4,6] => ? = 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [2,7,4,8,1,3,5,6] => ? = 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8,7,4,1,2,3,5,6] => ? = 0
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,3,7,8,1,4,5,6] => ? = 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,8,7,1,3,4,5,6] => ? = 0
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? = 0
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 0
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,3,1,2,4,5,6,7] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000366
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 0
[1,1] => [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 0
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,4,5,2] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,2,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,2,5,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,3,4,5,2] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,5,6,1,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,4,1,5,6,3] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,4,6,1,3,5] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,4,2,6,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,5,2,6,1,3] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,5,6,2] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,2,4,5] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,3,2,4,6,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,4,5,6,1,3] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,1,5,6,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,3,5] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,3,4,2,6,5] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,3,4,5,6,2] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [2,3,4,6,7,1,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [2,3,5,1,6,7,4] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [2,3,5,7,1,4,6] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [2,4,1,5,3,7,6] => ? = 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [3,5,6,2,7,1,4] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [2,4,1,5,6,7,3] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [2,4,7,1,3,5,6] => ? = 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,4,2,5,7,6] => ? = 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [4,2,5,6,7,1,3] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [4,5,2,1,6,7,3] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [4,5,2,7,1,3,6] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,4,5,2,7,6] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [4,6,2,3,7,1,5] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,4,5,6,7,2] => ? = 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,3,2,4,5,7,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,4,3,6,7,1,5] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,4,5,1,6,7,3] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,4,5,7,1,3,6] => ? = 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [4,2,1,5,3,7,6] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [5,3,6,2,7,1,4] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [4,2,1,5,6,7,3] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,7,1,3,5,6] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,3,4,2,5,7,6] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,4,5,6,7,1,3] => ? = 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [5,2,3,1,6,7,4] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,7,1,4,6] => ? = 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,3,4,5,2,7,6] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,3,4,5,6,7,2] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [2,3,4,5,7,8,1,6] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [2,3,4,6,1,7,8,5] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [2,3,4,6,8,1,5,7] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [2,3,5,1,6,4,8,7] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [3,4,6,7,2,8,1,5] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [2,3,5,1,6,7,8,4] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [2,3,5,8,1,4,6,7] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [2,4,1,5,3,6,8,7] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [3,5,2,6,7,8,1,4] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [3,5,6,2,1,7,8,4] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [3,5,6,2,8,1,4,7] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [2,4,1,5,6,3,8,7] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [3,5,7,2,4,8,1,6] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [2,4,1,5,6,7,8,3] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [2,4,8,1,3,5,6,7] => ? = 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [3,1,4,2,5,6,8,7] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [4,2,5,3,7,8,1,6] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [4,2,5,6,1,7,8,3] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [4,2,5,6,8,1,3,7] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [4,5,2,1,6,3,8,7] => ? = 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [5,6,3,7,2,8,1,4] => ? = 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [4,5,2,1,6,7,8,3] => ? = 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [4,5,2,8,1,3,6,7] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => [3,1,4,5,2,6,8,7] => ? = 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => [4,2,5,6,7,8,1,3] => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => [4,6,2,3,1,7,8,5] => ? = 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => 0
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => 0
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001086
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [2,1] => [1,2] => 0
[1,1] => [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[2] => [1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,4,1,2,5] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,5,1,2,6] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,4,6,2,1,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,4,1,2,5,6] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,5,2,6,1,4] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,2,1,4,6] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,2,4,1,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,2,4,5,6] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [4,2,5,6,1,3] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,2,5,1,3,6] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,3,1,5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,1,3,5,6] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,2,3,6,1,4] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,1,4,6] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6,2,3,4,1,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [3,4,5,6,7,1,2] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [3,4,5,6,1,2,7] => ? = 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [3,4,5,7,2,1,6] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [3,4,5,1,2,6,7] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [3,4,6,2,7,1,5] => ? = 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [3,4,6,2,1,5,7] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [3,4,7,2,5,1,6] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [3,4,1,2,5,6,7] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,5,2,6,7,1,4] => ? = 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,5,2,6,1,4,7] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,5,2,7,4,1,6] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,5,2,1,4,6,7] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,6,2,4,7,1,5] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,6,2,4,1,5,7] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,7,2,4,5,1,6] => ? = 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,2,4,5,6,7] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [4,2,5,6,7,1,3] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [4,2,5,6,1,3,7] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [4,2,5,7,3,1,6] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [4,2,5,1,3,6,7] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [4,2,6,3,7,1,5] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [4,2,6,3,1,5,7] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [4,2,7,3,5,1,6] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,1,3,5,6,7] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [5,2,3,6,7,1,4] => ? = 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [5,2,3,6,1,4,7] => ? = 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [5,2,3,7,4,1,6] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,1,4,6,7] => ? = 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [6,2,3,4,7,1,5] => ? = 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,1,5,7] => ? = 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [3,4,5,6,7,8,1,2] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [3,4,5,6,7,1,2,8] => ? = 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [3,4,5,6,8,2,1,7] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [3,4,5,6,1,2,7,8] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [3,4,5,7,2,8,1,6] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [3,4,5,7,2,1,6,8] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [3,4,5,8,2,6,1,7] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [3,4,5,1,2,6,7,8] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [3,4,6,2,7,8,1,5] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [3,4,6,2,7,1,5,8] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [3,4,6,2,8,5,1,7] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [3,4,6,2,1,5,7,8] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [3,4,7,2,5,8,1,6] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [3,4,7,2,5,1,6,8] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [3,4,8,2,5,6,1,7] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [3,4,1,2,5,6,7,8] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [3,5,2,6,7,8,1,4] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [3,5,2,6,7,1,4,8] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [3,5,2,6,8,4,1,7] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [3,5,2,6,1,4,7,8] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [3,5,2,7,4,8,1,6] => ? = 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [3,5,2,7,4,1,6,8] => ? = 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [3,5,2,8,4,6,1,7] => ? = 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [3,5,2,1,4,6,7,8] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => [3,6,2,4,7,8,1,5] => ? = 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => [3,6,2,4,7,1,5,8] => ? = 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => [4,2,5,1,3,6,7,8] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => 0
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,3] => [3,4,1,2,5,6,7,8,9] => 0
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5,1,3,6,7,8,9,4] => [4,2,5,1,3,6,7,8,9] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => 0
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,10,3] => [3,4,1,2,5,6,7,8,9,10] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [1,0]
=> [2,1] => 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 0 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 0 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? = 0 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 1 = 0 + 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 0 + 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,2] => ? = 0 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? = 0 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? = 0 + 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ? = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,2] => ? = 0 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => ? = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [8,6,4,5,1,7,2,3] => ? = 0 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => ? = 0 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => ? = 0 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,7,5,6,2,8,4] => ? = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? = 0 + 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 0 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,2] => ? = 0 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => ? = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [8,5,4,1,6,7,2,3] => ? = 1 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ? = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,5,4,1,6,2,8,3] => ? = 0 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => ? = 0 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [8,3,5,1,6,7,2,4] => ? = 0 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ? = 0 + 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 1 = 0 + 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 1 = 0 + 1
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1 = 0 + 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 1 = 0 + 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 1 = 0 + 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 1 = 0 + 1
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1 = 0 + 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000375
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 0
[1,1] => [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0]
=> [2,3,1] => [2,3,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,2,4,5,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,2,3,5,6,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,2,4,5,6,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,6,3,2,4,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,6,2,4] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,1,3,4,2,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,1,3,6,4,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,6,2,1,3,5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,3,1,4,5] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6,2,3,4,1,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,2,3,4,6,7,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,2,3,5,6,7,4] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,4,2,7,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,2,6,7,3,5] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,1,2,4,5,3,6] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,2,4,5,6,7,3] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,3,2,7,4,5,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,6,3,7,2,4,5] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,5,1,7,2,4,6] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,5,6,7,2,4] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,7,1,4,2,5,6] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [6,1,3,4,7,2,5] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,1,3,4,5,2,6] => ? = 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,1,3,4,7,5,6] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,1,3,6,7,4,5] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,5,1,7,3,4,6] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,1,5,6,7,3,4] => ? = 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,4,2,1,3,5,6] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [4,6,2,7,1,3,5] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [4,7,2,5,1,3,6] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,3,1,7,4,5,6] => ? = 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,6,3,7,1,4,5] => ? = 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [5,7,2,3,1,4,6] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,6,7,1,4] => ? = 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,3,4,1,5,6] => ? = 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => ? = 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => ? = 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [1,2,3,4,5,7,8,6] => ? = 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,2,3,4,8,5,7] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [1,2,3,4,6,7,8,5] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [1,5,2,3,8,4,6,7] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [5,1,2,3,7,8,4,6] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [8,1,2,3,5,6,4,7] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,2,3,5,6,7,8,4] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,2,8,4,3,5,6,7] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,4,2,7,8,3,5,6] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [4,6,1,2,8,3,5,7] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [4,1,2,6,7,8,3,5] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [4,8,1,2,5,3,6,7] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [7,1,2,4,5,8,3,6] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [8,1,2,4,5,6,3,7] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,2,4,5,6,7,8,3] => ? = 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [1,3,2,4,8,5,6,7] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [1,3,2,7,8,4,5,6] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [6,3,1,8,2,4,5,7] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [1,6,3,7,8,2,4,5] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [8,3,1,5,2,4,6,7] => ? = 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [3,5,1,7,8,2,4,6] => ? = 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [3,8,1,5,6,2,4,7] => ? = 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [3,1,5,6,7,8,2,4] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => [1,3,8,4,2,5,6,7] => ? = 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],2)
=> ? = 0
[1,1] => ([(0,1)],2)
=> ([(1,2)],3)
=> 0
[2] => ([],2)
=> ([],3)
=> ? = 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,2] => ([(1,2)],3)
=> ([(2,3)],4)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[3] => ([],3)
=> ([],4)
=> ? = 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,3] => ([(2,3)],4)
=> ([(3,4)],5)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[4] => ([],4)
=> ([],5)
=> ? = 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4] => ([(3,4)],5)
=> ([(4,5)],6)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[5] => ([],5)
=> ([],6)
=> ? = 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(4,5),(4,6),(5,6)],7)
=> 0
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,5] => ([(4,5)],6)
=> ([(5,6)],7)
=> 0
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4] => ([(3,5),(4,5)],6)
=> ([(4,6),(5,6)],7)
=> 0
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(3,6),(4,6),(5,6)],7)
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[6] => ([],6)
=> ([],7)
=> ? = 0
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001344
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
[2] => [1,1,0,0]
=> [2,3,1] => [2,1,3] => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,1,4,3,5] => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,2,3,6,5] => 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,4,2,6,5] => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,3,4,6,1,5] => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ? = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,5,2,3,4,7,6] => ? = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,2,3,6,7,4] => ? = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,2,4,3,5,7,6] => 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,6,2,3,7,5] => ? = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,5,1,2,3,7,6] => ? = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,2,5,6,7,3] => ? = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,3,4,5,2,7,6] => 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,2,6,4,7,5] => ? = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [1,3,2,5,4,7,6] => 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,5,6,2,7,4] => ? = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,4,5,1,2,7,6] => ? = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,4,2,6,7,5] => ? = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,4,1,2,5,7,6] => ? = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => ? = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,1,3,6,7,4,5] => ? = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,3,1,5,7,4,6] => ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,1,5,6,7,3,4] => ? = 0 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,4,5,1,7,3,6] => ? = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,1,4,3,6,5,7] => ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,4,1,3,7,5,6] => ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,1,4,5,6,3,7] => ? = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,3,4,7,1,5,6] => ? = 0 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,1,6,7,3,4,5] => ? = 0 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,5,1,7,3,4,6] => ? = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,1,3,5,4,6,7] => ? = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,3,7,1,4,5,6] => ? = 0 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,1,6,3,4,5,7] => ? = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? = 0 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 0 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => ? = 0 + 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,8,6] => ? = 0 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,6,2,3,4,5,8,7] => ? = 0 + 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,2,3,4,7,8,5] => ? = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [1,2,5,3,4,6,8,7] => ? = 0 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [5,1,7,2,3,4,8,6] => ? = 0 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [5,6,1,2,3,4,8,7] => ? = 0 + 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [5,1,2,3,6,7,8,4] => ? = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,2,4,5,3,6,8,7] => ? = 0 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [4,1,2,7,3,5,8,6] => ? = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,4,2,6,3,5,8,7] => ? = 0 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [4,1,2,6,3,7,8,5] => ? = 0 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [1,4,5,2,3,6,8,7] => ? = 0 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [4,1,5,2,3,7,8,6] => ? = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [4,5,1,2,3,6,8,7] => ? = 0 + 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [4,1,2,5,6,7,8,3] => ? = 0 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [1,3,4,5,6,2,8,7] => ? = 0 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [3,1,2,4,7,5,8,6] => ? = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [1,3,2,4,6,5,8,7] => ? = 1 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [3,1,2,6,7,4,8,5] => ? = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [1,3,5,2,6,4,8,7] => ? = 0 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [3,1,5,2,7,4,8,6] => ? = 0 + 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St000516
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 0
[1,1] => [1,0,1,0]
=> [3,1,2] => [3,2,1] => 0
[2] => [1,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [3,4,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,4,1] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,2,3,4,5,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,5,1] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,2,3,6,1,4] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,2,3,4,5,1] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,2,6,1,5,3] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,2,6,1,5,3] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,2,3,6,1,4] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,3,4,5,1] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,6,1,4,5,2] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,1,4,5,2] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,5,1,6,2,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,6,1,4,5,2] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [4,2,6,1,5,3] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [4,2,6,1,5,3] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,2,3,6,1,4] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => ? = 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,2,3,4,5,6,1] => ? = 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [6,2,3,4,7,1,5] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [7,2,3,4,5,6,1] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [5,2,3,7,1,6,4] => ? = 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [5,2,3,7,1,6,4] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [6,2,3,4,7,1,5] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [7,2,3,4,5,6,1] => ? = 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [4,2,7,1,5,6,3] => ? = 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [4,2,7,1,5,6,3] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [4,2,7,1,5,6,3] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [5,2,3,7,1,6,4] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [5,2,3,7,1,6,4] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [6,2,3,4,7,1,5] => ? = 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [7,2,3,4,5,6,1] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [3,7,1,4,5,6,2] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [3,7,1,4,5,6,2] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [3,6,1,4,7,2,5] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [3,7,1,4,5,6,2] => ? = 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [3,5,1,7,2,6,4] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [3,6,1,4,7,2,5] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [3,7,1,4,5,6,2] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [4,2,7,1,5,6,3] => ? = 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [4,2,7,1,5,6,3] => ? = 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [4,2,7,1,5,6,3] => ? = 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [5,2,3,7,1,6,4] => ? = 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [5,2,3,7,1,6,4] => ? = 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,2,3,4,7,1,5] => ? = 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,2,3,4,5,6,7,1] => ? = 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [8,2,3,4,5,6,7,1] => ? = 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [7,2,3,4,5,8,1,6] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [8,2,3,4,5,6,7,1] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [6,2,3,4,8,1,7,5] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [6,2,3,4,8,1,7,5] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [7,2,3,4,5,8,1,6] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [8,2,3,4,5,6,7,1] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [5,2,3,8,1,6,7,4] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [5,2,3,8,1,6,7,4] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [5,2,3,7,1,8,4,6] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [5,2,3,8,1,6,7,4] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [6,2,3,4,8,1,7,5] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [6,2,3,4,8,1,7,5] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [7,2,3,4,5,8,1,6] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [8,2,3,4,5,6,7,1] => ? = 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [4,2,8,1,5,6,7,3] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [4,2,8,1,5,6,7,3] => ? = 1
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000750
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000750: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000750: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [1,0]
=> [2,1] => 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? = 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ? = 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ? = 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? = 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,2] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ? = 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,2] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [6,4,1,5,2,7,8,3] => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [6,5,4,1,2,7,8,3] => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [6,1,5,2,3,7,8,4] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? = 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,2] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? = 1
Description
The number of occurrences of the pattern 4213 in a permutation.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset.
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