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Your data matches 78 different statistics following compositions of up to 3 maps.
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Matching statistic: St000031
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [4,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [2,1,4,7,5,8,3,6] => ?
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [1,3,2,5,7,6,8,4] => ?
=> ? = 4
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,4,5,2,3,6,7,8] => [1,3,5,4,2,6,7,8] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,6,2,3,5,7,8] => [1,3,5,4,6,2,7,8] => ?
=> ? = 5
[1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,4,7,2,3,5,8,6] => [1,3,5,4,7,8,2,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6,8] => [1,3,5,4,6,7,2,8] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => [1,3,5,4,6,7,8,2] => ?
=> ? = 3
[1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,5,2,3,6,8,4,7] => [1,3,5,7,2,6,8,4] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [1,3,4,5,7,6,8,2] => ?
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,8,4,5,7] => [2,3,6,5,7,1,8,4] => ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [4,3,1,2,7,5,8,6] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [7,4,3,1,2,5,8,6] => ?
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [8,2,4,6,1,3,5,7] => ?
=> ? = 6
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [3,1,4,2,8,7,5,6] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,5,1,2,6,8,4,7] => [3,1,8,7,4,2,5,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,6,2,3,5,7,8] => [1,4,2,6,5,3,7,8] => ?
=> ? = 5
[1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,4,7,2,3,5,8,6] => [1,4,2,8,6,5,3,7] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6,8] => [1,4,2,7,6,5,3,8] => ?
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [1,5,2,3,7,4,8,6] => [1,8,6,4,3,2,5,7] => ?
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6,8] => [1,7,6,4,3,2,5,8] => ?
=> ? = 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [11,10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,2,5,6,7,8,9] => [4,2,1,3,5,6,7,8,9] => ?
=> ? = 7
[1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,1,6,2,7,8,9] => [6,2,4,1,3,5,7,8,9] => ?
=> ? = 7
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [6,1,7,2,8,3,9,4,10,5] => [10,5,8,4,2,1,6,3,7,9] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [7,1,8,2,9,3,10,4,11,5,12,6] => [12,6,3,8,4,2,1,7,10,5,9,11] => ?
=> ? = 6
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,6,7,1,8,2,9,3] => [7,2,5,1,4,9,3,6,8] => ?
=> ? = 6
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [5,1,6,2,7,3,8,4,9] => [6,3,8,4,2,1,5,7,9] => ?
=> ? = 5
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000676
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 85%●distinct values known / distinct values provided: 73%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 85%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,7,4,8,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,5,1,2,6,8,4,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [3,1,6,2,7,4,8,5] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [4,1,5,2,6,3,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,1,5,2,6,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [4,1,5,2,7,3,8,6] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [4,5,6,1,2,8,3,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,8,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,6,2,7,3,8,5] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4,1,2,3,8,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,6,2,7,3,8,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,8,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,8,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [[.,.],[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ? = 10
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,9,1,4,5,6,7,8] => [[.,.],[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[]
=> [] => .
=> ?
=> ? = 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,10,9] => [[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8,10] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],[.,.]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000011
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 91%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,8,7] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,6,7,5,8] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,5,2,6,7,4,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,2,6,4,5,7,8] => [.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ?
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,7,5,8,6] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,6,1,7,2,8,5] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,5,1,2,6,8,4,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6,8] => [.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,6,7,2,3,5,8] => [.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,6,2,3,5,7,8] => [.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,4,2,3,5,7,6,8] => [.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,4,2,3,7,5,6,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,4,7,2,3,5,8,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => [.,[[.,[.,.]],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [4,1,5,2,6,3,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [4,1,5,2,7,3,8,6] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [4,5,6,1,2,8,3,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,8,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,2,3,5,4,8,6,7] => [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [1,2,5,3,6,4,7,8] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [1,5,2,3,6,7,4,8] => [.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,5,2,3,6,8,4,7] => [.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,5,2,3,6,4,7,8] => [.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,5,2,3,4,7,6,8] => [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000340
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 76%●distinct values known / distinct values provided: 73%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 76%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,3,6,7,8] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [2,5,1,6,3,7,4,8] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ?
=> ? = 4 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,4,5,8,6,7] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,7,5,8,6] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,2,5,6,7,8] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [1,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,4,5,2,3,8,6,7] => [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
[1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,4,2,3,5,8,6,7] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 - 1
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [4,1,5,2,6,3,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,1,5,2,6,3,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,2,3,5,4,8,6,7] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,5,2,3,6,8,4,7] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [1,5,2,3,7,4,8,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> [1,5,2,3,4,8,6,7] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,5,2,3,8,4,6,7] => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,2,3,6,8,4,5,7] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000245
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => [3,4,2,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [4,2,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [2,3,5,4,1] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [2,4,5,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [2,5,3,4,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [2,4,3,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => [4,5,2,3,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [3,4,1,5,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,2,3,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [3,5,4,2,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [5,4,2,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => [2,3,4,7,6,5,1] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,6,5,4,1,2,3] => [2,3,7,6,5,4,1] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [5,3,1,2,4,6,7] => [2,4,5,3,1,6,7] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [5,3,1,2,7,4,6] => [2,6,7,4,5,3,1] => ? = 4 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 3 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => [2,7,6,5,4,3,1] => ? = 2 - 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4,2,5,6,7] => [4,2,1,3,5,6,7] => [3,4,2,1,5,6,7] => ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => [6,2,4,1,3,5,7] => [3,4,1,5,6,2,7] => ? = 5 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 4 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,6,3,7] => [4,2,1,6,3,5,7] => [5,6,3,4,2,1,7] => ? = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => [1,6,7,4,5,3,2] => ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3 - 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => [6,7,5,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [8,7,6,5,1,2,3,4] => [2,3,4,8,7,6,5,1] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [8,4,1,2,3,5,6,7] => [2,3,5,6,7,8,4,1] => ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => [3,1,2,4,8,7,6,5] => [2,3,1,4,8,7,6,5] => ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,1,4,5,6,7] => [8,7,6,5,4,1,2,3] => [2,3,8,7,6,5,4,1] => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? = 7 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [2,1,4,3,8,6,5,7] => [2,1,4,3,7,8,6,5] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,3,6,7,8] => [5,3,1,2,4,6,7,8] => [2,4,5,3,1,6,7,8] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [8,3,5,1,2,4,6,7] => [2,4,5,1,6,7,8,3] => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [4,3,1,2,7,5,8,6] => ? => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [2,1,6,4,3,5,8,7] => [2,1,5,6,4,3,8,7] => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [2,1,8,6,4,3,5,7] => [2,1,5,7,8,6,4,3] => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [2,1,5,3,8,6,4,7] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [2,5,1,6,3,7,4,8] => [5,3,1,2,7,4,6,8] => ? => ? = 5 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [7,4,3,1,2,5,8,6] => ? => ? = 4 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,3,8,7,6,5,4] => [2,1,3,8,7,6,5,4] => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [8,7,6,5,4,3,1,2] => [2,8,7,6,5,4,3,1] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [1,3,2,5,4,8,7,6] => ? => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,5,2,6,7,4,8] => [1,7,4,2,3,5,6,8] => [1,3,5,6,7,4,2,8] => ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [1,3,2,6,4,7,5,8] => [1,3,2,7,5,6,4,8] => ? = 5 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [1,3,2,6,4,8,7,5] => ? => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,2,6,4,5,7,8] => [1,3,2,6,5,4,7,8] => ? => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => [1,3,2,8,6,5,4,7] => ? => ? = 4 - 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [1,3,2,7,6,5,4,8] => ? => ? = 4 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,4,5,8,6,7] => [1,3,2,4,5,8,7,6] => ? => ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [1,3,2,8,7,6,5,4] => [1,3,2,8,7,6,5,4] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [4,2,1,3,6,5,8,7] => [3,4,2,1,6,5,8,7] => ? = 4 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,2,5,6,7,8] => [4,2,1,3,5,6,7,8] => ? => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => [6,2,4,1,3,5,7,8] => ? => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [8,2,4,6,1,3,5,7] => [3,5,6,1,4,7,8,2] => ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [3,1,4,2,8,7,5,6] => ? => ? = 4 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,6,1,7,2,8,5] => [6,2,4,1,3,8,5,7] => [3,4,1,7,8,5,6,2] => ? = 5 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [3,1,4,2,8,7,6,5] => [4,2,3,1,8,7,6,5] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [6,4,2,1,3,5,8,7] => [3,5,6,4,2,1,8,7] => ? = 4 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000069
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 82%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [2,1] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [4,7,3,6,1,5,2] => ([(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6)],7)
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [7,6,5,1,8,4,3,2] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [4,8,7,6,1,5,3,2] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => [7,6,5,8,4,1,3,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,1,4,5,6,7] => [7,6,5,4,1,8,3,2] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2] => ([(5,6),(5,7)],8)
=> ? = 7
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [6,8,5,7,3,4,1,2] => ([(0,7),(1,5),(2,4),(3,6),(3,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [7,6,5,8,3,4,1,2] => ([(0,7),(1,7),(2,7),(3,6),(4,5)],8)
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,3,6,7,8] => [8,7,6,3,5,1,4,2] => ([(3,6),(3,7),(4,5),(4,7)],8)
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [3,8,7,1,6,5,4,2] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7)],8)
=> ? = 6
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [6,5,8,7,3,1,4,2] => ([(0,7),(1,5),(1,6),(2,5),(2,6),(3,4),(3,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => ([(0,7),(1,5),(2,4),(3,6),(3,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [6,8,4,7,3,5,1,2] => ([(0,6),(1,6),(1,7),(2,5),(3,4),(3,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [6,8,4,3,7,5,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(3,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [2,5,1,6,3,7,4,8] => [8,4,7,3,6,1,5,2] => ([(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [6,4,8,7,3,1,5,2] => ([(0,7),(1,5),(1,6),(2,5),(2,6),(2,7),(3,4),(3,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [5,8,4,7,3,6,1,2] => ([(0,7),(1,6),(1,7),(2,5),(3,4),(3,6),(3,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,8,7] => [7,8,5,4,3,6,1,2] => ([(0,7),(1,7),(2,7),(3,6),(4,5)],8)
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [7,6,5,4,8,3,1,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6)],8)
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [7,6,5,4,3,8,1,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6)],8)
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [7,6,5,4,3,1,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 7
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => ([(2,7),(3,6),(4,5)],8)
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [7,6,8,4,5,2,3,1] => ([(1,7),(2,7),(3,6),(4,5)],8)
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,5,2,6,7,4,8] => [8,4,7,6,2,5,3,1] => ([(2,6),(2,7),(3,4),(3,5),(3,7)],8)
=> ? = 6
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [8,5,4,7,6,2,3,1] => ?
=> ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [7,5,4,8,6,2,3,1] => ?
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,2,6,4,5,7,8] => [8,7,5,4,6,2,3,1] => ([(3,7),(4,7),(5,6)],8)
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => [6,8,5,4,7,2,3,1] => ([(1,7),(2,7),(3,6),(4,5),(4,7)],8)
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [8,6,5,4,7,2,3,1] => ([(2,7),(3,7),(4,7),(5,6)],8)
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,4,5,8,6,7] => [7,6,8,5,4,2,3,1] => ([(3,7),(4,7),(5,6)],8)
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [7,6,5,4,8,2,3,1] => ([(1,7),(2,7),(3,7),(4,7),(5,6)],8)
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [8,7,6,5,4,2,3,1] => ([(6,7)],8)
=> ? = 7
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [7,8,5,6,2,4,1,3] => ([(0,7),(1,5),(2,4),(3,6),(3,7)],8)
=> ? = 4
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,2,5,6,7,8] => [8,7,6,5,2,4,1,3] => ([(4,7),(5,6),(5,7)],8)
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => [8,7,2,6,1,5,4,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [2,8,1,7,6,5,4,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 6
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [7,5,8,6,2,1,4,3] => ([(0,7),(1,5),(1,6),(2,5),(2,6),(3,4),(3,7)],8)
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,6,1,7,2,8,5] => [5,8,2,7,1,6,4,3] => ([(0,4),(0,5),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,6),(2,7)],8)
=> ? = 5
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [7,6,5,8,2,1,4,3] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6)],8)
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [7,8,4,6,2,5,1,3] => ([(0,6),(1,6),(1,7),(2,5),(3,4),(3,7)],8)
=> ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,7,4,8,6] => [6,8,4,7,2,5,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,5,1,2,6,8,4,7] => [7,4,8,6,2,1,5,3] => ([(0,6),(1,5),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [3,1,6,2,7,4,8,5] => [5,8,4,7,2,6,1,3] => ([(0,5),(1,5),(1,7),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 4
[1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,4,5,6,2,7,3,8] => [8,3,7,2,6,5,4,1] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 6
[1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6,8] => [8,6,7,3,2,5,4,1] => ?
=> ? = 5
[1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,4,5,2,3,8,6,7] => [7,6,8,3,2,5,4,1] => ([(1,5),(2,5),(3,6),(3,7),(4,6),(4,7)],8)
=> ? = 4
[1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,6,7,2,3,5,8] => [8,5,3,2,7,6,4,1] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 5
[1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [7,5,3,2,8,6,4,1] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,6,2,3,5,7,8] => [8,7,5,3,2,6,4,1] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 5
Description
The number of maximal elements of a poset.
Matching statistic: St000007
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ? = 3
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ? = 6
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,6,2,4,5,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,2,7,3,5,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,6,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 4
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 3
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => ? = 4
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => ? = 3
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => ? = 3
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,1,2,3,6,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ? = 6
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 6
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,4,8,7,6,5,3,2] => ? = 6
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,1,4,5,6,7] => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,7,6,5,4,8,3,2] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,3,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => ? = 6
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,4,3,7,8,6,5,2] => ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,4,3,6,8,7,5,2] => ? = 4
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,4,3,6,8,7,5,2] => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,4,3,6,8,7,5,2] => ? = 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,5,4,3,8,7,6,2] => ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,8,7] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => ? = 3
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,7,6,5,4,8,3,2] => ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,6,7,5,8] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,8] => [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => ? = 5
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [2,4,7,6,8,5,3,1] => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => ? = 6
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,5,2,6,7,4,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [2,5,4,8,7,6,3,1] => ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [2,5,4,7,8,6,3,1] => ? = 4
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,2,6,4,5,7,8] => [.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [2,5,4,8,7,6,3,1] => ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ? => ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> [2,6,5,4,8,7,3,1] => ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,4,5,8,6,7] => [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [2,7,6,8,5,4,3,1] => ? = 5
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [2,7,6,5,4,8,3,1] => ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? = 7
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [2,1,5,7,8,6,4,3] => ? = 4
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: St000925
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,4,5},{2},{3}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,3,5},{2},{4}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,3,4,5},{2}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,2,5},{3},{4}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,2,4,5},{3}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,4,5},{2,3}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [4,2,3,1,8,6,7,5] => {{1,4},{2},{3},{5,8},{6},{7}}
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => {{1,5,6,7,8},{2},{3},{4}}
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [8,2,3,1,5,6,7,4] => {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => {{1,3},{2},{4},{5},{6,8},{7}}
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => [3,2,1,4,8,5,6,7] => {{1,3},{2},{4},{5,6,7,8}}
=> ? = 4
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [2,1,4,3,8,5,7,6] => {{1,2},{3,4},{5,6,8},{7}}
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [2,1,4,3,8,5,6,7] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,3,6,7,8] => [5,2,1,4,3,6,7,8] => {{1,3,5},{2},{4},{6},{7},{8}}
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [8,2,5,4,3,6,7,1] => {{1,8},{2},{3,5},{4},{6},{7}}
=> ? = 6
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [4,2,1,3,8,7,6,5] => {{1,3,4},{2},{5,8},{6,7}}
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [2,1,6,3,5,4,8,7] => {{1,2},{3,4,6},{5},{7,8}}
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [2,1,8,3,5,4,7,6] => {{1,2},{3,4,6,8},{5},{7}}
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [2,1,8,5,4,3,7,6] => {{1,2},{3,6,8},{4,5},{7}}
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [2,5,1,6,3,7,4,8] => [7,2,1,5,4,6,3,8] => {{1,3,7},{2},{4,5},{6},{8}}
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [8,2,1,3,5,7,6,4] => ?
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [2,1,8,3,6,5,7,4] => {{1,2},{3,4,8},{5,6},{7}}
=> ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,8,7] => [2,1,6,3,4,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,3,8,4,5,6,7] => {{1,2},{3},{4,5,6,7,8}}
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,5,4,8,6,7] => [1,3,2,5,4,8,6,7] => {{1},{2,3},{4,5},{6,7,8}}
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 6
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,5,2,6,7,4,8] => [1,7,3,2,5,6,4,8] => {{1},{2,4,7},{3},{5},{6},{8}}
=> ? = 6
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [1,3,2,7,6,5,4,8] => {{1},{2,3},{4,7},{5,6},{8}}
=> ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => [1,3,2,8,6,5,4,7] => {{1},{2,3},{4,7,8},{5,6}}
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,2,6,4,5,7,8] => [1,3,2,6,4,5,7,8] => {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,2,7,4,5,8,6] => ? => ?
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,5,6,7},{8}}
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,4,5,8,6,7] => [1,3,2,4,5,8,6,7] => {{1},{2,3},{4},{5},{6,7,8}}
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [1,3,2,8,4,5,6,7] => {{1},{2,3},{4,5,6,7,8}}
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [4,1,3,2,6,5,8,7] => {{1,2,4},{3},{5,6},{7,8}}
=> ? = 4
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,7,5,8,6] => [4,1,3,2,8,5,7,6] => {{1,2,4},{3},{5,6,8},{7}}
=> ? = 4
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,2,5,6,7,8] => [4,1,3,2,5,6,7,8] => {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => [6,4,3,2,5,1,7,8] => {{1,6},{2,4},{3},{5},{7},{8}}
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [8,6,3,4,5,2,7,1] => {{1,8},{2,6},{3},{4},{5},{7}}
=> ? = 6
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,8,5,7] => [4,3,2,1,8,6,5,7] => {{1,4},{2,3},{5,7,8},{6}}
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,6,1,7,2,8,5] => [8,4,3,2,6,5,7,1] => {{1,8},{2,4},{3},{5,6},{7}}
=> ? = 5
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [4,3,2,1,8,5,6,7] => {{1,4},{2,3},{5,6,7,8}}
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [6,1,3,2,5,4,8,7] => {{1,2,4,6},{3},{5},{7,8}}
=> ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,7,4,8,6] => [8,1,3,2,5,4,7,6] => {{1,2,4,6,8},{3},{5},{7}}
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,5,1,2,6,8,4,7] => [8,3,2,1,5,6,4,7] => {{1,4,7,8},{2,3},{5},{6}}
=> ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [3,1,6,2,7,4,8,5] => [8,1,3,2,6,5,7,4] => {{1,2,4,8},{3},{5,6},{7}}
=> ? = 4
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,1,2,4,5,8,6,7] => [3,1,2,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
The following 68 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000546The number of global descents of a permutation. St000167The number of leaves of an ordered tree. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000105The number of blocks in the set partition. St000024The number of double up and double down steps of a Dyck path. St000211The rank of the set partition. St000482The (zero)-forcing number of a graph. St000068The number of minimal elements in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000702The number of weak deficiencies of a permutation. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000332The positive inversions of an alternating sign matrix. St000308The height of the tree associated to a permutation. St000157The number of descents of a standard tableau. St001461The number of topologically connected components of the chord diagram of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000542The number of left-to-right-minima of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000239The number of small weak excedances. St000288The number of ones in a binary word. St000325The width of the tree associated to a permutation. St000389The number of runs of ones of odd length in a binary word. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001083The number of boxed occurrences of 132 in a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001820The size of the image of the pop stack sorting operator. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001863The number of weak excedances of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St000782The indicator function of whether a given perfect matching is an L & P matching. St001935The number of ascents in a parking function.
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