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Your data matches 108 different statistics following compositions of up to 3 maps.
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Matching statistic: St000011
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 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.
Matching statistic: St000382
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2] => 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [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
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,9] => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,7] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,3,1,5] => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,1,6] => ? = 1
[]
=> []
=> []
=> [] => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ?
=> ? => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,7] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,6] => ? = 3
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 11
[1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,1,1,1,1,3] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,1,1,5] => ? = 2
[1,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,1,1,1,1,4] => ? = 2
[1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,1,6] => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ?
=> ? => ? = 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> ?
=> ? => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? => ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0]
=> ?
=> ? => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,9,1] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000288
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,1] => 100000001 => ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,1] => 100000001 => ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,1,1] => 100000011 => ? = 3
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,1,1] => 100000011 => ? = 3
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,1,1,1] => 100000111 => ? = 4
[1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,1,1,1] => 100000111 => ? = 4
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ? => ? = 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1] => 100001111 => ? = 5
[1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1] => 100001111 => ? = 5
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => 100011111 => ? = 6
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => 100011111 => ? = 6
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1] => 100111111 => ? = 7
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1] => 100111111 => ? = 7
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => 101111111 => ? = 8
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? => ? => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 8
[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? => ? => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 9
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? => ? => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? => ? => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ? = 1
[]
=> []
=> [] => ? => ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => 110111111 => ? = 8
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => 110011111 => ? = 7
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => 110011111 => ? = 7
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => 111011111 => ? = 8
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => 111101111 => ? = 8
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ? => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,2,1] => 1101010101 => ? = 6
[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? => ? => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1] => 101011111 => ? = 7
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => 100011111 => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => 100011111 => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => 100011111 => ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,1,1] => 100000111 => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => 100000001 => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,6] => 1000100000 => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 7
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,4] => 1000001000 => ? = 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => 1100000000 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,9] => 1100000000 => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000097
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [11] => ([],11)
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11] => ([],11)
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4
[1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6
[1,0,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 8
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 1
[]
=> []
=> [] => ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,2,2,2] => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 8
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001581
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 88%●distinct values known / distinct values provided: 58%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 88%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [11] => ([],11)
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St001462
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [[1,2],[3,4],[5]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => [[1,2],[3,4],[5]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,8,3,7] => [5,7,3,4,1,8,2,6] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4,8] => [4,7,6,1,5,3,2,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,7,3,8,4] => [3,8,1,6,5,4,7,2] => ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,5,6,7,4,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,6,1,7,3,4,8,5] => [3,8,1,6,5,4,7,2] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,6,7,1,3,4,5,8] => [4,7,6,1,5,3,2,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7,8] => [5,2,3,6,1,4,7,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => [5,2,3,7,1,6,4,8] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,3,4,7,1,8,5,6] => [5,2,3,8,1,7,6,4] => ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6,8] => [5,2,3,7,1,6,4,8] => ?
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,7,1,8,6] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,3,4,5,8,1,6,7] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,7,2,5,8] => [6,3,2,7,5,1,4,8] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,7,2,8,5,6] => [5,2,3,8,1,7,6,4] => ?
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,4,5,7,2,8,6] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,4,5,8,2,6,7] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,5,7,4,8,6] => [3,2,1,6,8,4,7,5] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,5,2,7,8,4,6] => [1,4,7,2,8,6,3,5] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,2,5,8,4,6,7] => [3,2,1,6,8,4,7,5] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [3,1,2,5,6,7,4,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,6,4,7,5,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,3,4,2,6,7,8,5] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,3,4,2,7,5,8,6] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,3,4,2,8,5,6,7] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3,7,8] => [6,2,5,4,3,1,7,8] => ?
=> ? = 3
[1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,7,3,8,6] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [4,1,2,5,8,3,6,7] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,4,5,6,2,7,3,8] => [1,7,5,4,3,6,2,8] => ?
=> ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,6,3,7,5,8] => [5,2,3,7,1,6,4,8] => ?
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,8,5,7] => [5,7,3,4,1,8,2,6] => ?
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5,7,8] => [5,2,3,6,1,4,7,8] => ?
=> ? = 3
[1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,2,3,6,7,8,5] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,7,5,8,6] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,1,2,7,3,5,6,8] => [5,2,3,7,1,6,4,8] => ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,4,2,3,8,5,6,7] => [1,4,3,2,8,6,7,5] => ?
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4,7,8] => [6,2,5,4,3,1,7,8] => ?
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,8,6] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,3,8,4,6,7] => [6,2,3,4,8,1,7,5] => ?
=> ? = 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,6,2,3,7,8,4,5] => [1,8,3,4,7,6,5,2] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,7,2,3,8,4,5,6] => [1,8,3,4,7,6,5,2] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [9,2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2],[9]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [9,2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2],[9]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [10,2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [8,2,3,4,5,6,7,1,9] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [8,2,3,4,5,6,7,1,9] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10,2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [11,2,3,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9,10],[2],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [9,2,3,4,5,6,7,8,1,10] => [[1,3,4,5,6,7,8,10],[2],[9]]
=> ? = 2
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 92%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 5 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000306
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [5,4,2,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,1,4,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,5,3,4,7,6] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,1,4,6,7] => [5,3,2,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [3,2,1,4,7,5,6] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,4,1,5,6,7] => [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,5,2,6,7] => [5,4,3,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [3,4,1,5,2,6,7] => [5,4,1,3,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,4,5,1,2,6,7] => [5,1,4,3,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,7,6] => [3,1,2,5,4,7,6] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,5,2,4,6,7] => [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [3,5,1,2,4,6,7] => [5,1,3,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,5,4,6,7] => [3,1,2,5,4,6,7] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5,7] => [3,1,2,4,6,5,7] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5,7] => [1,4,3,2,6,5,7] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,7,5] => [3,1,2,4,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,2,4,7,5,6] => [3,1,2,4,7,5,6] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,4,2,5,7,6] => [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => [5,1,2,4,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3,6,7] => [5,1,4,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6,7] => [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5,7] => [1,4,2,3,6,5,7] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,2,3,5,7,6] => [1,4,2,3,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000678
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000675
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
St000675: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000925The number of topologically connected components of a set partition. St000098The chromatic number of a graph. St000010The length of the partition. St000383The last part of an integer composition. St001050The number of terminal closers of a set partition. St000971The smallest closer of a set partition. St000069The number of maximal elements of a poset. St000504The cardinality of the first block of a set partition. St000502The number of successions of a set partitions. St000025The number of initial rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001494The Alon-Tarsi number of a graph. St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St000234The number of global ascents of a permutation. St000068The number of minimal elements in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000054The first entry of the permutation. St000237The number of small exceedances. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000553The number of blocks of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St001330The hat guessing number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000297The number of leading ones in a binary word. St000740The last entry of a permutation. St000738The first entry in the last row of a standard tableau. St000203The number of external nodes of a binary tree. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000286The number of connected components of the complement of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama algebra. St000883The number of longest increasing subsequences of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St001545The second Elser number of a connected graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000924The number of topologically connected components of a perfect matching. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001889The size of the connectivity set of a signed permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000942The number of critical left to right maxima of the parking functions.
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