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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000674
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 2
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [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] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => [3,2,1,5,4,6,7,8] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,1,4,6,7,8,5] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => [3,2,1,4,6,5,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,1,4,5,7,6,8] => [3,2,1,4,5,7,6,8] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,1,4,7,5,8,6] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,4,7,8,5,6] => [3,2,1,4,8,5,7,6] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,1,4,7,5,6,8] => [3,2,1,4,7,5,6,8] => ?
=> ? = 2
[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] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,6,3,7,8] => ? => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ? => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6,8] => ? => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6,8] => ? => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,4,5,3,8,6,7] => ? => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,6,7,5,8] => [2,1,4,3,7,6,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,6,3,7,5,8] => [2,1,7,6,4,3,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,6,7,3,5,8] => [2,1,7,4,3,6,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,4,3,5,7,6,8] => ? => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,4,3,7,5,6,8] => ? => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => ? => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6,8] => ? => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,4,8,3,5,6,7] => [2,1,8,4,3,5,6,7] => ?
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,5,6,7,4,8] => ? => ?
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ? => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6,8] => ? => ?
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,7,4,8,6] => ? => ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,7,8,4,6] => ? => ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6,8] => ? => ?
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,1,3,5,4,8,6,7] => ? => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,1,5,3,6,8,4,7] => [2,1,8,6,5,3,4,7] => ?
=> ? = 0
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,6,4,7,8] => ? => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [2,1,5,6,3,7,4,8] => [2,1,7,6,3,5,4,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [2,1,5,6,7,3,4,8] => [2,1,7,3,6,5,4,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,1,5,6,3,8,4,7] => ? => ?
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,1,5,6,8,3,4,7] => [2,1,8,3,6,5,4,7] => ?
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,1,5,6,3,4,7,8] => [2,1,6,3,5,4,7,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,7,8,6] => ? => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6,8] => ? => ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ? => ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6,8] => [2,1,7,3,5,4,6,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [2,1,5,3,4,8,6,7] => ? => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,1,5,8,3,4,6,7] => [2,1,8,3,5,4,6,7] => ?
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,1,3,4,6,7,5,8] => [2,1,3,4,7,6,5,8] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [2,1,3,4,6,8,5,7] => ? => ?
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [2,1,3,6,4,7,8,5] => ? => ?
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [2,1,3,6,4,7,5,8] => ? => ?
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [2,1,3,6,7,4,5,8] => ? => ?
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [2,1,3,6,8,4,5,7] => ? => ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,1,6,3,4,7,5,8] => ? => ?
=> ? = 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000247
(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
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 89%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 89%
Values
[1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,2,3}}
=> 0
[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] => {{1,2,3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> 0
[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] => {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => {{1,2,3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,1,4,5,7,6,8] => {{1,2,3},{4},{5},{6,7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,1,4,7,5,8,6] => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,4,7,8,5,6] => {{1,2,3},{4},{5,7},{6,8}}
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,1,4,7,5,6,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,1,4,5,6,8,7] => {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 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] => {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => {{1,2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,6,3,7,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,7,8,3,6] => {{1,2},{3,4,5,7},{6,8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => {{1,2},{3,4,5},{6},{7,8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,4,5,3,8,6,7] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,4,5,3,6,7,8] => {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,6,7,5,8] => {{1,2},{3,4},{5,6,7},{8}}
=> ? = 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] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,7,8] => {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,6,3,7,5,8] => {{1,2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => {{1,2},{3,4,6},{5,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,6,7,3,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => {{1,2},{3,4,6},{5,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,6,3,5,7,8] => {{1,2},{3,4,5,6},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => {{1,2},{3,4},{5},{6,7,8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => {{1,2},{3,4},{5,7},{6,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,3,5,6,8,7] => {{1,2},{3,4},{5},{6},{7,8}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,4,3,5,8,6,7] => {{1,2},{3,4},{5},{6,7,8}}
=> ? = 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] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8] => {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,8,4] => {{1,2},{3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,5,6,7,4,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,6,4,8,7] => {{1,2},{3},{4,5,6},{7,8}}
=> ? = 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => {{1,2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[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] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[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] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => {{1,2,3},{4,5},{6},{7},{8}}
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> {{1,5,6},{2},{3},{4},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => {{1,2,3},{4},{5,6},{7,8}}
=> {{1,3,5,6},{2},{4},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,4,5,6},{2},{3},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,1,4,5,7,6,8] => {{1,2,3},{4},{5},{6,7},{8}}
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,1,4,7,5,8,6] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,4,7,8,5,6] => {{1,2,3},{4},{5,7},{6,8}}
=> {{1,3,5,6},{2,4},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,1,4,7,5,6,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,1,4,5,6,8,7] => {{1,2,3},{4},{5},{6},{7,8}}
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,4,5,6},{2},{3},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,1,4,8,5,6,7] => {{1,2,3},{4},{5,6,7,8}}
=> {{1,5,6},{2},{3},{4},{7},{8}}
=> ? = 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] => {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => {{1,2},{3,4,5,6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,6,8,3,7] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,6,3,7,8] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,7,3,8,6] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,7,8,3,6] => {{1,2},{3,4,5,7},{6,8}}
=> {{1,3,7},{2,4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => {{1,2},{3,4,5},{6},{7,8}}
=> {{1,3,4,7},{2},{5},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,4,5,3,8,6,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,4,5,8,3,6,7] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,4,5,3,6,7,8] => {{1,2},{3,4,5},{6},{7},{8}}
=> {{1,2,3,4,7},{5},{6},{8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,6,7,5,8] => {{1,2},{3,4},{5,6,7},{8}}
=> {{1,2,5,7},{3},{4},{6},{8}}
=> ? = 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] => {{1,2},{3,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,7,8] => {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,6,3,7,8,5] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,6,3,7,5,8] => {{1,2},{3,4,5,6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,4,7},{2},{3,5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,6,7,8,3,5] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,6,7,3,5,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,8,5,7] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,4,7},{2},{3,5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,6,3,5,7,8] => {{1,2},{3,4,5,6},{7},{8}}
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => {{1,2},{3,4},{5},{6,7,8}}
=> {{1,4,5,7},{2},{3},{6},{8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,4,3,5,7,6,8] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => {{1,2},{3,4},{5,7},{6,8}}
=> {{1,3,5,7},{2,4},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => ?
=> ?
=> ? = 0
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000011
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,1,2,8,6,7,3,5] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,5,2,8,7,4,6] => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,1,4,2,3,8,5,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [6,1,4,2,3,7,8,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [6,1,5,2,3,7,8,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,6,5,2,8,4,7] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,5,2,7,8,4] => [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,4,7,6,2,8,5] => [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,7,5,6,2,8,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,8,3,7,5,6] => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,6,1,5,3,8,4,7] => [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
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: St000288
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 00 => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 11 => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 000 => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 001 => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 100 => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 000 => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0001 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 0000 => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0000 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0011 => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1000 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1001 => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 0000 => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0000 => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 0001 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1100 => 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0000 => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => 00000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 00000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => 00011 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => 00000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => 00001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => 00000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => 00000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => 00001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 00100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => 00000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => 00000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 00111 => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 10000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 10001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 10000 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => 10000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => 10011 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 00000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => 00001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 00000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => 00000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => 00000 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => 00000 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => 00000 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => 00011 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 11000 => 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [3,4,5,1,7,2,6,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,5,7,1,8,2,6] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [3,4,5,1,2,8,6,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [3,4,5,1,8,2,6,7] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [3,4,1,6,8,2,5,7] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,1,6,2,5,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,6,1,7,2,5,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,8,5,7] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,1,8,2,5,7] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,6,1,2,5,7,8] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,4,1,2,7,5,6,8] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,4,1,7,2,8,5,6] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,4,1,7,8,2,5,6] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,4,1,7,2,5,6,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,4,7,1,2,8,5,6] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,4,1,2,5,8,6,7] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [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
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,4,8,1,2,5,6,7] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,1,5,6,7,2,4,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,5,2,4,6,7,8] => ? => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,5,1,6,2,8,4,7] => ? => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [3,5,6,1,7,2,4,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,7,1,2,4,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,5,6,1,2,4,7,8] => ? => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,7,1,2,8,4,6] => ? => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,7,1,8,2,4,6] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,5,8,1,2,4,6,7] => ? => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [3,1,2,6,7,4,5,8] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [3,1,2,6,4,8,5,7] => ? => ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [3,1,2,6,8,4,5,7] => ? => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,1,2,6,4,5,7,8] => ? => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [3,1,6,2,7,8,4,5] => ? => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [3,1,6,7,8,2,4,5] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [3,6,1,2,7,8,4,5] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [3,6,1,2,7,4,5,8] => ? => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,6,1,7,2,4,5,8] => ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [3,6,7,1,8,2,4,5] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [3,6,1,2,8,4,5,7] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,6,8,1,2,4,5,7] => ? => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [3,1,2,7,8,4,5,6] => ? => ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [3,1,7,2,4,8,5,6] => ? => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [3,1,7,2,8,4,5,6] => ? => ? = 0
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,7,1,8,2,4,5,6] => ? => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,4,5,6,2,8,3,7] => ? => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3,7,8] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,4,5,7,2,8,3,6] => ? => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,4,5,2,8,3,6,7] => ? => ? = 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ? => ? = 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000678
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 89%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 89%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000312
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 89%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 89%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,3,4] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,1,5,7,3,4,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0
Description
The number of leaves in a graph.
That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000439
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
Description
The position of the first down step of a Dyck path.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000675The number of centered multitunnels of a Dyck path. St000022The number of fixed points of a permutation. St000925The number of topologically connected components of a set partition. St000215The number of adjacencies of a permutation, zero appended. St000895The number of ones on the main diagonal of an alternating sign matrix. St000502The number of successions of a set partitions. St000237The number of small exceedances. St000260The radius of a connected graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000221The number of strong fixed points of a permutation. St000117The number of centered tunnels of a Dyck path. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St000164The number of short pairs. St001479The number of bridges of a graph. St000025The number of initial rises of a Dyck path. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000717The number of ordinal summands of a poset. St000214The number of adjacencies of a permutation. St001545The second Elser number of a connected graph. St000894The trace of an alternating sign matrix. St000843The decomposition number of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000239The number of small weak excedances. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000654The first descent of a permutation. St000153The number of adjacent cycles of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000075The orbit size of a standard tableau under promotion. St001903The number of fixed points of a parking function. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000732The number of double deficiencies of a permutation. St000989The number of final rises of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001637The number of (upper) dissectors of a poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001810The number of fixed points of a permutation smaller than its largest moved point.
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