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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000647
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St001085
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,4,3,2,1,6,8,7] => [5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,8] => [5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [5,4,3,2,1,7,8,6] => [5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,8,5,4,3,2,1] => [2,3,1,4,5,6,7,8] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,10,9] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,10,9,8,7] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [6,5,4,3,2,1,9,8,7,10] => [6,5,4,3,2,1,10,9,8,7] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,6,5,7,4,8,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,7,6,8,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,4,7,6,8,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,8,7,9,6,10,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,8,7,9,6,10,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,5,8,7,9,4,10,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,7,6,8,5,9,4,10,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,8,10,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,9,8,10,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,5,7,4,9,8,10,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,6,9,8,10,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,7,6,9,8,10,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,6,5,10,9,11,8,12,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,4,6,3,10,9,11,8,12,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,5,7,4,10,9,11,8,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,7,6,10,9,11,8,12,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,4,7,6,10,9,11,8,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,5,8,7,10,9,11,4,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [2,1,7,6,8,5,10,9,11,4,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [2,1,6,5,9,8,10,7,11,4,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [2,1,7,6,9,8,10,5,11,4,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,7,10,9,11,6,12,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,4,8,7,10,9,11,6,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [2,1,8,7,9,6,10,5,11,4,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,9,8,10,7,11,6,12,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [2,1,5,4,9,8,10,7,11,6,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,8,7,11,10,12,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3,8,7,11,10,12,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,6,5,7,4,8,3,11,10,12,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5,11,10,12,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,7,6,8,3,11,10,12,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,8,7,9,6,11,10,12,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,8,7,9,6,11,10,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [2,1,6,5,8,7,9,4,11,10,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [2,1,7,6,8,5,9,4,11,10,12,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,8,11,10,12,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3,9,8,11,10,12,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001083
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,6,8,7] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,5,7,6,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,5,7,8,6] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,4,6,7,8,5] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,2,1,4,6,8,7,5] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,2,1,4,7,6,5,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [3,2,1,4,7,6,8,5] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,1,4,7,8,6,5] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,2,1,4,8,7,6,5] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,7,6,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [3,2,1,5,6,4,7,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [3,2,1,5,6,7,4,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,8,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,5,6,8,7,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> [3,2,1,5,7,6,4,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [3,2,1,5,7,6,8,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,1,5,7,8,6,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,1,5,8,7,6,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,2,1,6,5,7,4,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [3,2,1,6,5,7,8,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,2,1,6,7,5,4,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,2,1,6,7,5,8,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,6,7,8,5,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,2,1,6,8,7,5,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,2,1,7,6,5,4,8] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,2,1,7,6,5,8,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [3,2,1,7,6,8,5,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,1,7,8,6,5,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => ? = 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [6,5,7,8,4,3,2,1] => [6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,5,8,7,4,3,2,1] => [6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,7,8,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8,9,10] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5,10,9,8] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5,10,9,8] => [4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,8,7,6,5,9,10] => [4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => ? = 0
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000386
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => [8,7,6,4,3,5,1,2] => ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,3,2,1] => [8,7,6,4,2,3,5,1] => ?
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => [8,7,5,6,4,1,2,3] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => [8,7,4,5,3,2,6,1] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => [8,7,4,5,3,1,2,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => [8,7,4,5,2,1,3,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => [8,7,4,3,1,2,5,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => [8,7,4,2,3,5,1,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => [8,7,4,2,3,1,5,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => [8,6,7,5,4,1,2,3] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,2,3,1] => [8,6,7,4,5,2,1,3] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => [8,6,7,4,3,5,1,2] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => [8,6,7,4,3,1,2,5] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,2,3,1] => [8,6,7,4,2,3,5,1] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,8,2,3,1] => [8,6,7,4,2,3,1,5] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => [8,6,7,4,1,2,3,5] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,2,3,1] => [8,6,7,3,4,2,1,5] => ?
=> ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,2,4,1] => [8,6,5,7,4,2,1,3] => ?
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,3,2,5,1] => [8,6,5,4,7,2,1,3] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,2,5,1] => [8,6,4,5,7,2,1,3] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,2,7,1] => [8,6,4,5,2,3,7,1] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,2,8,1] => [8,6,4,5,2,3,1,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,2,8,1] => [8,6,4,5,2,1,3,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,6,1] => [8,6,3,4,5,7,1,2] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => [8,6,3,4,2,5,7,1] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,2,8,1] => [8,6,3,4,2,5,1,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ?
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,2,3,4,1] => [8,5,6,7,4,2,1,3] => ?
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,2,3,4,1] => [8,5,6,7,3,1,2,4] => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,2,3,6,1] => [8,5,6,3,4,7,2,1] => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,2,3,8,1] => [8,5,6,3,2,4,1,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,2,4,8,1] => [8,5,4,6,3,1,2,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,2,4,8,1] => [8,5,4,6,2,1,3,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => [8,5,4,6,1,2,3,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => [8,5,3,4,2,6,1,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => [8,5,3,2,4,1,6,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [8,7,5,2,3,4,6,1] => [8,4,5,6,3,7,2,1] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [6,7,5,2,3,4,8,1] => [8,4,5,6,3,1,2,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,2,3,5,6,1] => [8,4,5,3,6,7,2,1] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,2,3,6,8,1] => [8,4,5,3,2,6,1,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,7,8,1] => [8,4,5,3,1,2,6,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => [8,4,5,2,1,3,6,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,4,3,2,5,6,8,1] => [8,4,3,2,5,6,1,7] => ?
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [8,3,2,4,1,5,6,7] => ?
=> ? = 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000292
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> [1] => => ? = 0
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0 => 0
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => 1 => 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => 10 => 0
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 100 => 0
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 100 => 0
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 110 => 0
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 1
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0010 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1100 => 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [5,4,7,8,6,3,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [5,6,4,8,7,3,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [5,4,6,8,7,3,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [3,6,7,8,5,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [3,7,6,5,8,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [3,6,7,5,8,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [3,5,7,6,8,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [4,3,6,8,7,5,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
=> [4,3,7,6,8,5,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [4,5,3,8,7,6,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [4,6,5,3,8,7,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [4,7,6,5,3,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [4,6,7,5,3,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [5,4,6,3,8,7,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [3,4,7,6,8,5,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [3,6,5,4,8,7,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [3,6,7,5,4,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [3,5,7,6,4,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [3,6,5,7,4,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [3,5,6,7,4,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [.,[.,[[[.,[.,.]],.],[.,[.,.]]]]]
=> [4,3,5,8,7,6,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [4,6,5,3,7,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [3,4,6,7,5,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [3,5,6,4,7,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],[.,.]],.]]]
=> [3,4,5,7,6,8,2,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [3,5,4,6,7,8,2,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [2,6,5,8,7,4,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
=> [2,7,6,5,8,4,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,[[.,[[.,.],.]],.]]]]
=> [2,6,7,5,8,4,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[.,[[[.,.],[.,.]],.]]]]
=> [2,5,7,6,8,4,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [2,4,7,6,8,5,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [2,5,4,7,8,6,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [2,6,7,5,4,8,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [2,5,7,6,4,8,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
=> [2,6,5,7,4,8,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [2,5,6,7,4,8,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [2,5,6,4,7,8,3,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [2,4,5,7,6,8,3,1] => ? => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> [3,2,7,8,6,5,4,1] => ? => ? = 0
Description
The number of ascents of a binary word.
Matching statistic: St000291
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 00 => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 00 => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 01 => 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 10 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 001 => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 001 => 0
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 011 => 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 110 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 0000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0001 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 0001 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0001 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0001 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0001 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0011 => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [8,6,3,5,4,7,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [8,4,3,7,6,5,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,7,5,6,4,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [8,3,7,4,5,6,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [8,4,3,6,5,7,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [8,3,6,4,5,7,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [8,4,3,5,7,6,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [8,3,4,7,6,5,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [8,3,4,7,5,6,2,1] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [8,3,4,6,5,7,2,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [8,6,3,5,4,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [8,5,3,4,6,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [8,4,3,6,5,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [8,3,6,4,5,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [8,3,5,4,6,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [8,3,4,6,5,2,7,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [8,4,3,5,2,7,6,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [8,3,5,4,2,7,6,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [8,3,4,5,2,7,6,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [8,4,3,2,7,6,5,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [8,3,4,2,7,6,5,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [8,4,3,2,6,7,5,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [8,3,2,5,7,6,4,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [8,2,4,7,5,6,3,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [8,2,4,5,7,6,3,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [8,2,4,5,6,7,3,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [8,3,5,4,2,6,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [8,3,4,5,2,6,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [8,3,2,6,5,4,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> [8,2,5,4,6,3,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [8,2,4,6,5,3,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [8,2,4,5,6,3,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [8,4,3,2,5,7,6,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [8,3,2,4,7,6,5,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [8,2,4,3,7,6,5,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [8,2,3,7,6,5,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [8,2,3,6,5,7,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [8,2,3,5,6,7,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [8,2,4,5,3,6,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [8,2,3,5,6,4,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [8,3,2,4,5,7,6,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [8,2,3,4,7,6,5,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [8,2,4,3,5,6,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [8,2,3,4,6,5,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [8,2,3,4,5,7,6,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [7,3,6,4,5,2,8,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [7,3,5,4,6,2,8,1] => ? => ? = 0
Description
The number of descents of a binary word.
Matching statistic: St000007
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,4,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,3,4,2,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [7,6,4,3,5,1,2] => [1,2,3,5,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [7,6,4,3,2,5,1] => [1,2,5,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => [1,2,5,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [7,6,4,2,3,5,1] => [1,2,3,5,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => [1,3,5,2,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => [1,2,3,5,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => [1,2,5,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => [1,2,5,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => [1,2,4,5,3,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => [1,4,5,2,3,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => [1,2,4,5,3,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [7,6,2,3,1,4,5] => [1,4,5,2,3,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [7,5,6,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [7,5,6,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [7,5,6,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [7,5,6,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [7,5,4,6,2,1,3] => [1,3,2,4,6,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [7,5,4,3,6,1,2] => [1,2,3,6,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [7,5,4,3,2,6,1] => [1,2,6,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [7,5,4,3,1,2,6] => [1,2,6,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [7,5,4,2,3,6,1] => [1,2,3,6,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [7,5,4,2,1,3,6] => [1,3,6,2,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [7,5,4,1,2,3,6] => [1,2,3,6,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [7,5,3,4,2,6,1] => [1,2,6,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [7,5,3,4,1,2,6] => [1,2,6,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => [7,5,3,2,4,6,1] => [1,2,4,6,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [7,5,3,2,1,4,6] => [1,4,6,2,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [7,5,3,1,2,4,6] => [1,2,4,6,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [7,5,2,3,1,4,6] => [1,4,6,2,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,5,1] => [7,4,5,3,6,2,1] => [1,2,3,6,4,5,7] => ? = 0 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St001036
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,3,2,1] => [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,3,2,1] => [8,7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,3,2,1] => [8,7,6,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,8,4,3,2,1] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,4,3,2,1] => [8,7,6,5,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,2,1] => [8,7,6,5,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,4,3,2,1] => [8,7,6,5,2,3,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,3,2,1] => [8,7,6,4,5,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,3,2,1] => [8,7,6,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,2,1] => [8,7,6,4,5,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => [8,7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => [8,7,6,4,3,5,1,2] => ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,2,1] => [8,7,6,4,3,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,3,2,1] => [8,7,6,4,3,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,3,2,1] => [8,7,6,4,2,3,5,1] => ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,8,3,2,1] => [8,7,6,4,2,3,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,2,1] => [8,7,6,4,1,2,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => [8,7,6,3,4,5,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,7,3,2,1] => [8,7,6,3,4,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,3,2,1] => [8,7,6,3,4,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,8,3,2,1] => [8,7,6,3,4,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,3,2,1] => [8,7,6,3,2,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,3,2,1] => [8,7,6,3,2,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,3,2,1] => [8,7,6,3,2,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,3,2,1] => [8,7,6,3,1,2,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,2,1] => [8,7,6,2,3,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,3,2,1] => [8,7,6,2,3,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,2,1] => [8,7,6,2,3,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,3,2,1] => [8,7,6,2,1,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,4,2,1] => [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,4,2,1] => [8,7,5,6,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,4,2,1] => [8,7,5,6,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => [8,7,5,6,4,1,2,3] => ?
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,4,2,1] => [8,7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => [8,7,5,6,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [7,6,5,8,3,4,2,1] => [8,7,5,6,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,3,4,2,1] => [8,7,5,6,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000356
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1,4] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,1,3] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,1,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,1,2,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,1,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,2,1,3,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1,2,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,4,2,1,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,4,1,2,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,2,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,3,2,1,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,3,1,2,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,1,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,2,1,3,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,5,2,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,2,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,1,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,3,5,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [6,5,4,3,1,2,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [6,5,4,2,3,1,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [6,5,4,2,1,3,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [6,5,4,1,2,3,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [6,5,3,4,1,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [6,5,3,2,4,1,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [6,5,3,2,1,4,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [6,5,3,1,2,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [6,5,2,3,4,1,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [6,5,2,3,1,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,5,2,1,3,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [6,5,1,2,3,4,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [6,4,5,3,2,1,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [6,4,5,3,1,2,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,4,5,2,1,3,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [6,4,5,1,2,3,7] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [6,4,3,5,1,2,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [6,4,3,2,1,5,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,3,1,2,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [6,4,2,3,5,1,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [6,4,2,3,1,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [6,4,2,1,3,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [6,4,1,2,3,5,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [6,3,4,5,1,2,7] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [6,3,4,2,1,5,7] => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [6,3,4,1,2,5,7] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [6,3,2,4,5,1,7] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [6,3,2,4,1,5,7] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [6,3,2,1,4,5,7] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [6,3,1,2,4,5,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [6,2,3,4,5,1,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [6,2,3,4,1,5,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [6,2,3,1,4,5,7] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [6,2,1,3,4,5,7] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [5,6,4,3,1,2,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [5,6,4,2,3,1,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [5,6,4,2,1,3,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [5,6,4,1,2,3,7] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [5,6,3,4,2,1,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [5,6,3,2,4,1,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [5,6,3,2,1,4,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [5,6,3,1,2,4,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [5,6,2,3,4,1,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,6,2,3,1,4,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [5,6,2,1,3,4,7] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [5,4,6,3,1,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [5,4,6,2,3,1,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [5,4,6,2,1,3,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [5,4,6,1,2,3,7] => ? = 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => [3,2,1] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => [1,3,2] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,4,3] => [2,4,3,1] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [3,2,4,1] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,1,3,4] => [1,3,4,2] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,4,3,2] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,1,2,3] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4,1,3,2] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => [2,1,4,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => [4,2,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,5,4] => [2,3,5,4,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,4,3,5] => [2,4,3,5,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => [2,4,5,3,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => [2,5,4,3,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,2,4,5] => [3,2,4,5,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,3,2,5,4] => [3,2,5,4,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,4,2,3,5] => [3,4,2,5,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => [3,4,5,2,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => [3,5,4,2,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [4,3,2,5,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,5,3,2,4] => [4,3,5,2,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => [4,5,3,2,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,1,3,4,5] => [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [1,3,5,4,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => [1,4,3,5,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,1,5,3,4] => [1,4,5,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,5,4,3,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,4,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,5,4,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,3,5] => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,1,2,3,4] => [3,4,5,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [5,1,2,4,3] => [3,5,4,1,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,1,3,2,4] => [4,3,5,1,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [5,1,4,2,3] => [4,5,3,1,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,4,3,1,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [1,2,3,7,5,4,6] => [2,3,6,5,7,4,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,2,7,3,6,5,4] => [2,4,7,6,5,3,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => [2,5,4,3,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [1,2,6,4,3,5,7] => [2,5,4,6,3,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,2,7,4,3,5,6] => [2,5,4,6,7,3,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,2,7,4,3,6,5] => [2,5,4,7,6,3,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,2,6,5,3,4,7] => [2,5,6,4,3,7,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,2,7,5,3,4,6] => [2,5,6,4,7,3,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,2,7,6,3,4,5] => [2,5,6,7,4,3,1] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,2,7,6,3,5,4] => [2,5,7,6,4,3,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,2,6,5,4,3,7] => [2,6,5,4,3,7,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [1,2,7,5,4,3,6] => [2,6,5,4,7,3,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,2,7,6,4,3,5] => [2,6,5,7,4,3,1] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,2,7,6,5,3,4] => [2,6,7,5,4,3,1] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,2,7,6,5,4,3] => [2,7,6,5,4,3,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,3,2,4,7,5,6] => [3,2,4,6,7,5,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,3,2,4,7,6,5] => [3,2,4,7,6,5,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,3,2,5,4,6,7] => [3,2,5,4,6,7,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,3,2,5,4,7,6] => [3,2,5,4,7,6,1] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [1,3,2,6,4,5,7] => [3,2,5,6,4,7,1] => ? = 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000646The number of big ascents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000619The number of cyclic descents of a permutation. St000779The tier of a permutation. St000710The number of big deficiencies of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000451The length of the longest pattern of the form k 1 2. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000058The order of a permutation. St001862The number of crossings of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001728The number of invisible descents of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000534The number of 2-rises of a permutation. St001487The number of inner corners of a skew partition.
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