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Your data matches 96 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => [3]
=> 1
[1,1,0,0]
=> [2,3,1] => [3,2,1] => [2,1]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [4]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => [3,1]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => [4]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => [5]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => [3,2]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => [3,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => [5]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [2,2,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [4,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => [5]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => [5]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,4,5,2,1] => [3,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [3,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => [5]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => [6]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,6,3,4] => [4,2]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,2,3,5] => [3,3]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,3,2,4] => [6]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,2,5,3] => [3,2,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,6,1,2,4,5] => [4,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,1,6,2,4] => [2,2,2]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,4,2,1,3,5] => [6]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,5,2,3,1,4] => [6]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,4,6,1,2,3] => [4,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,1,4,2,5] => [3,2,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,5,1,2,6,4] => [4,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,5,2,1,4,3] => [6]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,4,5,2] => [2,2,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => [5,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,1,6,3,4] => [3,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,1,3,5] => [3,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,2,5,3,1,4] => [5,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,5,3] => [2,2,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [6]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,1,5,6,2,4] => [4,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [6]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => [5,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,1,5,6,3,2] => [4,2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,2,1,5] => [4,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,6,5,1,2,4] => [6]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,1,5,2,3] => [6]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,5,1] => [3,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000340
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 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]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 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]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,1,2,3,4,9,8,5,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [9,1,2,3,4,7,8,5,6] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,1,2,3,4,7,8,9,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,1,2,3,9,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [9,1,2,3,6,8,4,5,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,1,2,3,6,9,8,4,7] => [1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,8,5,9,7] => ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => [1,1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [8,1,2,5,9,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,9,8,3,6,7] => [1,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,9,2,4,5,6,7,8] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,8,2,4,5,6,9,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,9,8,2,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [3,1,4,8,9,2,5,6,7] => [1,1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,1,4,5,6,2,8,9,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [9,1,8,5,6,7,2,3,4] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,9,1,3,4,5,6,7,8] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [5,9,1,2,3,4,6,7,8] => [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [9,6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,9,1,4,5,6,7,8] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,9,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,8,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [9,5,4,1,2,3,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,9,1,5,6,7,8] => [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [2,3,9,8,1,4,5,6,7] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,9,1,6,7,8] => [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,7,4,5,6,1,2,3,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [2,3,9,5,6,7,1,4,8] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St001036
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 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]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 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]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,1,2,3,4,9,8,5,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [9,1,2,3,4,7,8,5,6] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,1,2,3,4,7,8,9,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,1,2,3,9,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [9,1,2,3,6,8,4,5,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,1,2,3,6,9,8,4,7] => [1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,8,5,9,7] => ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => [1,1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [8,1,2,5,9,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,9,8,3,6,7] => [1,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,9,2,4,5,6,7,8] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,8,2,4,5,6,9,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,9,8,2,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [3,1,4,8,9,2,5,6,7] => [1,1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,1,4,5,6,2,8,9,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [9,1,8,5,6,7,2,3,4] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,9,1,3,4,5,6,7,8] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [5,9,1,2,3,4,6,7,8] => [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [9,6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,9,1,4,5,6,7,8] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,9,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,8,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [9,5,4,1,2,3,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,9,1,5,6,7,8] => [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [2,3,9,8,1,4,5,6,7] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,9,1,6,7,8] => [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,7,4,5,6,1,2,3,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [2,3,9,5,6,7,1,4,8] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000159
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 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]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 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]
=> []
=> 0 = 1 - 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]
=> [4]
=> 1 = 2 - 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]
=> [3,3]
=> 1 = 2 - 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]
=> []
=> 0 = 1 - 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]
=> [4,3]
=> 2 = 3 - 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]
=> [2,2,2]
=> 1 = 2 - 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]
=> [4,2,2]
=> 2 = 3 - 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]
=> []
=> 0 = 1 - 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]
=> []
=> 0 = 1 - 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]
=> [4]
=> 1 = 2 - 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]
=> [3,3,2]
=> 2 = 3 - 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]
=> [2,2,2]
=> 1 = 2 - 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]
=> []
=> 0 = 1 - 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]
=> [4,3,2]
=> 3 = 4 - 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]
=> [1,1,1,1]
=> 1 = 2 - 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]
=> [4,1,1,1]
=> 2 = 3 - 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]
=> [3,3,1,1]
=> 2 = 3 - 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]
=> [1,1,1,1]
=> 1 = 2 - 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]
=> [4,3,1,1]
=> 3 = 4 - 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]
=> []
=> 0 = 1 - 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]
=> [4]
=> 1 = 2 - 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]
=> []
=> 0 = 1 - 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 = 2 - 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]
=> [4]
=> 1 = 2 - 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]
=> [3,3]
=> 1 = 2 - 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]
=> []
=> 0 = 1 - 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]
=> []
=> 0 = 1 - 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]
=> [4,3]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1]
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1]
=> ? = 4 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1]
=> ? = 5 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1]
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1]
=> ? = 5 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1]
=> ? = 5 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> ? = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 6 - 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]
=> [6,4,4]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3]
=> ? = 3 - 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]
=> [5,5,3,3]
=> ? = 3 - 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]
=> [6,5,3,3]
=> ? = 4 - 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]
=> [6,4,4,3]
=> ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 5 - 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]
=> [6,2,2,2,2]
=> ? = 3 - 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]
=> [5,5,2,2,2]
=> ? = 3 - 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]
=> [4,4,4,2,2]
=> ? = 3 - 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]
=> [6,4,4,2,2]
=> ? = 4 - 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]
=> [6,2,2,2,2]
=> ? = 3 - 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]
=> [6,4,4]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 4 - 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]
=> [6,3,3,3,2]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 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]
=> [6,2,2,2,2]
=> ? = 3 - 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]
=> [6,2,2,2,2]
=> ? = 3 - 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]
=> [6,5,2,2,2]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 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]
=> [6,3,3,3,2]
=> ? = 4 - 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]
=> [6,2,2,2,2]
=> ? = 3 - 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]
=> [5,5,4,3,2]
=> ? = 5 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2]
=> ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2]
=> ? = 6 - 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]
=> [5,5,1,1,1,1]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4,1,1,1]
=> ? = 3 - 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]
=> [6,4,4,1,1,1]
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [5,5,4,1,1,1]
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,1,1]
=> ? = 5 - 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]
=> [3,3,3,3,1,1]
=> ? = 3 - 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]
=> [6,3,3,3,1,1]
=> ? = 4 - 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]
=> [5,5,3,3,1,1]
=> ? = 4 - 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]
=> [3,3,3,3,1,1]
=> ? = 3 - 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 2
[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]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 4
[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
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 2
[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]
=> [3,3]
=> 2
[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
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[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]
=> [2,2,2]
=> 2
[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]
=> [4,2,2]
=> 3
[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
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 2
[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]
=> [3,3,2]
=> 3
[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]
=> [2,2,2]
=> 2
[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
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 4
[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]
=> [1,1,1,1]
=> 2
[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]
=> [4,1,1,1]
=> 3
[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]
=> [3,3,1,1]
=> 3
[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]
=> [1,1,1,1]
=> 2
[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]
=> [4,3,1,1]
=> 4
[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
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 2
[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
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 2
[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]
=> [4]
=> 2
[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]
=> [3,3]
=> 2
[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
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[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]
=> [6,5]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4]
=> ? = 2
[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]
=> [6,4,4]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 4
[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]
=> [3,3,3,3]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3]
=> ? = 3
[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]
=> [5,5,3,3]
=> ? = 3
[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]
=> [3,3,3,3]
=> ? = 2
[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]
=> [6,5,3,3]
=> ? = 4
[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]
=> [6,5]
=> ? = 3
[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]
=> [6,4,4,3]
=> ? = 4
[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]
=> [3,3,3,3]
=> ? = 2
[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]
=> [3,3,3,3]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3]
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 5
[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]
=> [6,2,2,2,2]
=> ? = 3
[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]
=> [5,5,2,2,2]
=> ? = 3
[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]
=> [4,4,4,2,2]
=> ? = 3
[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]
=> [6,4,4,2,2]
=> ? = 4
[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]
=> [6,2,2,2,2]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4]
=> ? = 2
[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]
=> [6,4,4]
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 4
[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]
=> [6,3,3,3,2]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3
[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]
=> [6,2,2,2,2]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2]
=> ? = 3
[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]
=> [6,2,2,2,2]
=> ? = 3
[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]
=> [6,5,2,2,2]
=> ? = 4
[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]
=> [6,5]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3
[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]
=> [6,3,3,3,2]
=> ? = 4
[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]
=> [6,2,2,2,2]
=> ? = 3
[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]
=> [5,5,4,3,2]
=> ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1,1,1,1]
=> ? = 3
[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]
=> [5,5,1,1,1,1]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4,1,1,1]
=> ? = 3
[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]
=> [6,4,4,1,1,1]
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,7,1,3,6,4,8,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1,1,1,1]
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [5,5,4,1,1,1]
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,1,1]
=> ? = 5
[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]
=> [3,3,3,3,1,1]
=> ? = 3
[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]
=> [6,3,3,3,1,1]
=> ? = 4
[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]
=> [5,5,3,3,1,1]
=> ? = 4
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 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]
=> 0 = 1 - 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 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 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 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 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 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 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 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 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]
=> 0 = 1 - 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 = 2 - 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 = 2 - 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]
=> 0 = 1 - 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]
=> 2 = 3 - 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 = 2 - 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]
=> 2 = 3 - 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]
=> 0 = 1 - 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]
=> 0 = 1 - 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 = 2 - 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]
=> 2 = 3 - 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 = 2 - 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]
=> 0 = 1 - 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]
=> 3 = 4 - 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 = 2 - 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]
=> 2 = 3 - 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]
=> 2 = 3 - 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 = 2 - 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]
=> 3 = 4 - 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]
=> 0 = 1 - 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 = 2 - 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]
=> 0 = 1 - 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 = 2 - 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 = 2 - 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 = 2 - 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]
=> 0 = 1 - 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]
=> 0 = 1 - 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]
=> 2 = 3 - 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]
=> ? = 2 - 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]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 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]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 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]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 3 - 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]
=> ? = 3 - 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]
=> ? = 2 - 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]
=> ? = 4 - 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]
=> ? = 2 - 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]
=> ? = 3 - 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]
=> ? = 4 - 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]
=> ? = 2 - 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]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 3 - 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]
=> ? = 3 - 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]
=> ? = 3 - 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]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,8,2,6,4,5,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,1,8,2,6,7,4,5] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 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]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 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]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 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]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,8,6,2,4,5,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 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]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,8,5,2,7,4,6] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 4 - 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]
=> ? = 2 - 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]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 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]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 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]
=> ? = 3 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000996
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2 = 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]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3 = 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,3,6,4,5] => 1 = 2 - 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]
=> [1,2,5,3,4,6] => 1 = 2 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,5,6,3,4] => 2 = 3 - 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]
=> [1,4,2,3,5,6] => 1 = 2 - 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]
=> [1,4,2,6,3,5] => 2 = 3 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,3,6,4,5] => 1 = 2 - 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]
=> [1,4,5,2,3,6] => 2 = 3 - 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]
=> [1,4,2,3,5,6] => 1 = 2 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,4,5,6,2,3] => 3 = 4 - 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]
=> [3,1,2,4,5,6] => 1 = 2 - 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]
=> [3,1,2,6,4,5] => 2 = 3 - 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]
=> [3,1,5,2,4,6] => 2 = 3 - 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]
=> [3,1,2,4,5,6] => 1 = 2 - 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]
=> [3,1,5,6,2,4] => 3 = 4 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,3,6,4,5] => 1 = 2 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [6,1,2,3,4,5] => 1 = 2 - 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]
=> [1,2,3,6,4,5] => 1 = 2 - 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]
=> [1,2,5,3,4,6] => 1 = 2 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,3,4,5,6] => 0 = 1 - 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]
=> [1,2,5,6,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,2,6,7,4,5] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,5,2,4,6,7] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,5,6,2,4,7] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,5,2,4,6,7] => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => ? = 5 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [3,4,1,2,7,5,6] => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5,7] => ? = 4 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [3,4,1,6,7,2,5] => ? = 5 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,2,6,7,4,5] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => ? = 5 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [3,4,1,2,7,5,6] => ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 4 - 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]
=> [1,2,3,4,7,5,6,8] => ? = 2 - 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]
=> [1,2,3,4,7,8,5,6] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ? = 4 - 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]
=> [1,2,5,3,4,6,7,8] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4,8,6,7] => ? = 3 - 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]
=> [1,2,5,3,7,4,6,8] => ? = 3 - 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]
=> [1,2,5,3,4,6,7,8] => ? = 2 - 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]
=> [1,2,5,3,7,8,4,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 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]
=> [1,2,3,4,7,5,6,8] => ? = 2 - 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]
=> [1,2,3,4,7,8,5,6] => ? = 3 - 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]
=> [1,2,5,6,3,8,4,7] => ? = 4 - 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]
=> [1,2,5,3,4,6,7,8] => ? = 2 - 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]
=> [1,2,5,3,4,6,7,8] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4,8,6,7] => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,8,3,4] => ? = 5 - 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]
=> [1,2,3,4,7,5,6,8] => ? = 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]
=> [1,2,3,4,7,5,6,8] => ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ? = 4 - 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]
=> [1,4,5,2,3,8,6,7] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,4,5,2,3,6,7,8] => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => ? = 3 - 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]
=> [1,4,2,3,7,8,5,6] => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 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]
=> [1,2,3,4,7,5,6,8] => ? = 2 - 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]
=> [1,2,3,4,7,8,5,6] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,4,5,2,3,6,7,8] => ? = 3 - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000695
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000695: Set partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000695: Set partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,2,3}}
=> 1
[1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => {{1,2,4},{3,5}}
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => {{1,3},{2,4,5}}
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,3,4,5},{2}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => {{1,4},{2},{3,5}}
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,4,5},{2},{3}}
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => {{1,3,4,5},{2}}
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => {{1,2,3,4,5}}
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => {{1,2,3,4,5,6}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,6,3,4] => {{1,2,3,5},{4,6}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,2,3,5] => {{1,2,4},{3,5,6}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,3,2,4] => {{1,2,3,4,5,6}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,2,5,3] => {{1,2,4},{3,6},{5}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,6,1,2,4,5] => {{1,3},{2,4,5,6}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,1,6,2,4] => {{1,3},{2,5},{4,6}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,4,2,1,3,5] => {{1,2,3,4,5,6}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,5,2,3,1,4] => {{1,2,3,4,5,6}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,4,6,1,2,3] => {{1,2,4,5},{3,6}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,1,4,2,5] => {{1,3},{2,5,6},{4}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,5,1,2,6,4] => {{1,3},{2,4,5,6}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,5,2,1,4,3] => {{1,2,3,4,5,6}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,4,5,2] => {{1,3},{2,6},{4},{5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => {{1,3,4,5,6},{2}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,1,6,3,4] => {{1,3,5},{2},{4,6}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,1,3,5] => {{1,4},{2},{3,5,6}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,2,5,3,1,4] => {{1,3,4,5,6},{2}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,5,3] => {{1,4},{2},{3,6},{5}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => {{1,2,3,4,5,6}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,1,5,6,2,4] => {{1,2,3,5},{4,6}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => {{1,2,3,4,5,6}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => {{1,2,3,4,6},{5}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,1,5,6,3,2] => {{1,2,4,6},{3,5}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,2,1,5] => {{1,3,5,6},{2,4}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,6,5,1,2,4] => {{1,2,3,4,5,6}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,1,5,2,3] => {{1,2,3,4,5,6}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,5,1] => {{1,3,6},{2,4},{5}}
=> 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [6,1,2,3,8,4,5,7] => {{1,2,3,4,6},{5,7,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,2,3,8,4,7,5] => {{1,2,3,4,6},{5,8},{7}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [5,1,2,8,3,4,6,7] => {{1,2,3,5},{4,6,7,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [5,1,2,7,3,8,4,6] => {{1,2,3,5},{4,7},{6,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [5,1,2,7,3,4,8,6] => {{1,2,3,5},{4,6,7,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [5,1,2,8,3,6,7,4] => {{1,2,3,5},{4,8},{6},{7}}
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [4,1,8,2,3,5,6,7] => {{1,2,4},{3,5,6,7,8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [4,1,7,2,3,8,5,6] => {{1,2,4},{3,5,7},{6,8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [4,1,6,2,8,3,5,7] => {{1,2,4},{3,6},{5,7,8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [4,1,8,2,7,5,3,6] => {{1,2,4},{3,5,6,7,8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [4,1,6,2,8,3,7,5] => {{1,2,4},{3,6},{5,8},{7}}
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,3,4,5,2,6] => {{1,2,7},{3,4,5,6,8}}
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [6,1,5,8,2,3,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => [6,1,5,8,2,3,7,4] => {{1,2,3,5,6},{4,8},{7}}
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [4,1,7,2,5,8,3,6] => {{1,2,4},{3,7},{5},{6,8}}
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => [4,1,6,2,3,8,5,7] => {{1,2,4},{3,5,6,7,8}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [4,1,7,2,3,5,8,6] => {{1,2,4},{3,5,6,7,8}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [4,1,6,2,7,8,5,3] => {{1,2,4},{3,6,8},{5,7}}
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [4,1,7,2,5,3,8,6] => {{1,2,4},{3,6,7,8},{5}}
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [4,1,8,2,5,6,7,3] => {{1,2,4},{3,8},{5},{6},{7}}
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [3,7,1,2,4,8,5,6] => {{1,3},{2,4,5,7},{6,8}}
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [3,6,1,2,8,4,5,7] => {{1,3},{2,4,6},{5,7,8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [3,5,1,8,2,4,6,7] => {{1,3},{2,5},{4,6,7,8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [3,5,1,7,2,8,4,6] => {{1,3},{2,5},{4,7},{6,8}}
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => [3,7,1,6,8,2,4,5] => {{1,3},{2,4,6,7},{5,8}}
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,1,4,2,3,8,5,7] => [6,4,2,1,8,3,5,7] => {{1,2,3,4,6},{5,7,8}}
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => [6,7,2,3,8,1,4,5] => {{1,6},{2,3,4,7},{5,8}}
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => [6,5,2,8,1,3,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [5,4,8,1,2,3,6,7] => {{1,2,4,5},{3,6,7,8}}
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => [5,4,7,1,2,8,3,6] => {{1,2,4,5},{3,7},{6,8}}
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [5,4,8,1,2,6,7,3] => {{1,2,4,5},{3,8},{6},{7}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => [3,7,1,4,2,8,5,6] => {{1,3},{2,5,7},{4},{6,8}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [3,8,1,4,7,5,2,6] => {{1,3},{2,5,6,7,8},{4}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => [3,5,1,2,7,8,4,6] => {{1,3},{2,4,5,7},{6,8}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [3,8,1,2,4,5,7,6] => {{1,3},{2,4,5,6,8},{7}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => [3,6,1,2,7,8,5,4] => {{1,3},{2,4,6,8},{5,7}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,5,2,7,8,4] => [3,5,1,6,8,4,7,2] => {{1,3},{2,5,8},{4,6},{7}}
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,8,2,1,3,5,4,6] => {{1,4,7},{2,3,5,6,8}}
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,5,2,6,8,1,4,3] => {{1,4,6,7},{2,3,5,8}}
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => [6,5,8,1,4,2,3,7] => {{1,2,4,5,6},{3,7,8}}
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,5,8,1,4,2,7,3] => {{1,2,4,5,6},{3,8},{7}}
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [3,6,1,4,2,8,5,7] => {{1,3},{2,5,6,7,8},{4}}
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [3,7,1,4,2,5,8,6] => {{1,3},{2,5,6,7,8},{4}}
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,4,7,6,2,8,5] => [3,6,1,4,7,8,5,2] => {{1,3},{2,6,8},{4},{5,7}}
=> ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,7,5,6,2,8,4] => [3,6,1,8,7,5,2,4] => {{1,3},{2,5,6,7},{4,8}}
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => [3,8,1,4,5,6,2,7] => {{1,3},{2,7,8},{4},{5},{6}}
=> ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [3,7,1,4,5,2,8,6] => {{1,3},{2,6,7,8},{4},{5}}
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,7,1,4,2,8,6,5] => {{1,3},{2,5,6,7,8},{4}}
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,8,1,4,5,6,7,2] => {{1,3},{2,8},{4},{5},{6},{7}}
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => [7,2,1,3,4,8,5,6] => {{1,3,4,5,7},{2},{6,8}}
=> ? = 3
Description
The number of blocks in the first part of the atomic decomposition of a set partition.
Let $\pi=(b_1,\dots,b_k)$ be a set partition with $k$ blocks, such that $\min b_i < \min b_{i+1}$. Then this statistic is the smallest number $\ell$ such that the union of the first $\ell$ blocks $b_1\cup\dots\cup b_\ell$ is an interval $\{1,\dots,m\}$.
The analogue for the decomposition of permutations is [[St000501]].
Matching statistic: St000925
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 4
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3,4,5}}
=> 2
[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]
=> {{1,5,6},{2,3,4}}
=> 2
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3,4},{5}}
=> 3
[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]
=> {{1,4,5,6},{2,3}}
=> 2
[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]
=> {{1,6},{2,3},{4,5}}
=> 3
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3,4,5}}
=> 2
[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]
=> {{1,5,6},{2,3},{4}}
=> 3
[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]
=> {{1,4,5,6},{2,3}}
=> 2
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3},{4},{5}}
=> 4
[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]
=> {{1,3,4,5,6},{2}}
=> 2
[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]
=> {{1,6},{2},{3,4,5}}
=> 3
[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]
=> {{1,5,6},{2},{3,4}}
=> 3
[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]
=> {{1,3,4,5,6},{2}}
=> 2
[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]
=> {{1,6},{2},{3,4},{5}}
=> 4
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3,4,5}}
=> 2
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,2,3,4,6},{5}}
=> 2
[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]
=> {{1,6},{2,3,4,5}}
=> 2
[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]
=> {{1,5,6},{2,3,4}}
=> 2
[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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,2,3,4,5,6}}
=> 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]
=> {{1,6},{2,3,4},{5}}
=> 3
[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]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[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]
=> {{1,8},{2,3,4,5,6},{7}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 2
[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]
=> {{1,8},{2,3,4,5},{6,7}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4,5},{6},{7}}
=> ? = 4
[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]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> {{1,8},{2,3,4},{5,6,7}}
=> ? = 3
[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]
=> {{1,7,8},{2,3,4},{5,6}}
=> ? = 3
[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]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
[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]
=> {{1,8},{2,3,4},{5,6},{7}}
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[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]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[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]
=> {{1,8},{2,3,4,5,6},{7}}
=> ? = 3
[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]
=> {{1,8},{2,3,4},{5},{6,7}}
=> ? = 4
[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]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
[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]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> {{1,8},{2,3,4},{5,6,7}}
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3,4},{5}}
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 5
[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]
=> {{1,8},{2,3},{4,5,6,7}}
=> ? = 3
[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]
=> {{1,7,8},{2,3},{4,5,6}}
=> ? = 3
[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]
=> {{1,6,7,8},{2,3},{4,5}}
=> ? = 3
[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]
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? = 4
[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]
=> {{1,8},{2,3},{4,5,6,7}}
=> ? = 3
[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]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[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]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 2
[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]
=> {{1,8},{2,3,4,5},{6,7}}
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4,5},{6},{7}}
=> ? = 4
[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]
=> {{1,8},{2,3},{4},{5,6,7}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 3
[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]
=> {{1,8},{2,3},{4,5,6,7}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,4,5,6,8},{2,3},{7}}
=> ? = 3
[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]
=> {{1,8},{2,3},{4,5,6,7}}
=> ? = 3
[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]
=> {{1,8},{2,3},{4,5,6},{7}}
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[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]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[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]
=> {{1,8},{2,3,4,5,6},{7}}
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 3
[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]
=> {{1,8},{2,3},{4},{5,6,7}}
=> ? = 4
[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]
=> {{1,8},{2,3},{4,5,6,7}}
=> ? = 3
[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]
=> {{1,7,8},{2,3},{4},{5},{6}}
=> ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3},{4},{5}}
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2},{3,4,5,6,7}}
=> ? = 3
[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]
=> {{1,7,8},{2},{3,4,5,6}}
=> ? = 3
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000011
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 90%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [[.,[.,[.,.]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [[.,[.,.]],[[.,[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [[.,[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [[.,[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [[.,[[.,[.,.]],.]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [[.,[.,.]],[[[.,.],.],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [[.,[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [[.,.],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[[.,[.,[.,.]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [[[.,[.,[.,.]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [[[.,[.,.]],.],[[.,.],.]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [[[.,[.,.]],[[.,.],.]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [[[.,[.,[.,.]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [[[.,[.,.]],.],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [[.,[.,[.,.]]],[[[.,[.,.]],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [[.,[.,[[.,[.,.]],.]]],[[.,.],.]]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => [[.,[.,[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [[.,[.,[.,.]]],[[[.,[.,.]],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => [[.,[.,[[.,[.,.]],[[.,.],.]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => [[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => [[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,1,4,2,3,8,5,7] => [[.,[[.,[.,.]],[.,.]]],[[.,.],.]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [8,1,4,2,3,7,5,6] => [[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => [[.,[[.,[.,[.,.]]],.]],[[.,.],.]]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => [[.,[[.,[.,[.,.]]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,1,4,2,6,3,5,7] => [[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [[.,[[.,[.,.]],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [8,1,5,2,7,3,4,6] => [[.,[[.,[.,[.,.]]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [8,1,4,2,6,7,3,5] => [[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => [[.,[.,.]],[.,[[[.,[.,.]],.],.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => [[.,[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => [[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [[.,[[.,[.,.]],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => [[.,[[.,[.,.]],[.,.]]],[[.,.],.]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => [[.,[[.,[.,.]],[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => [[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => [[.,[[[.,[.,[.,.]]],.],[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => [[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [[.,[.,.]],[.,[[[.,[.,.]],.],.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,4,7,6,2,8,5] => [[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => [[.,[[.,[.,.]],[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [[.,[[.,[.,.]],[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
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.
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000167The number of leaves of an ordered tree. St000065The number of entries equal to -1 in an alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000024The number of double up and double down steps of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000991The number of right-to-left minima of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000069The number of maximal elements of a poset. St000007The number of saliances of the permutation. St000711The number of big exceedences of a permutation. St000546The number of global descents of a permutation. St000068The number of minimal elements in a poset. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001889The size of the connectivity set of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St000989The number of final rises of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000942The number of critical left to right maxima of the parking functions. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000894The trace of an alternating sign matrix. St000075The orbit size of a standard tableau under promotion.
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