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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000022
(load all 91 compositions to match this statistic)
(load all 91 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000475
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[]
=> [] => ? => ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,1,2,1] => ?
=> ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11] => [11] => ?
=> ? = 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] => ?
=> ? = 0
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => [1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2,2] => [2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ?
=> ? = 4
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [12] => [12] => ?
=> ? = 0
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? => ? => ?
=> ? = 3
[1,1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000674
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
St000674: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> ? = 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 0
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 0
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 0
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000247
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 73%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 2
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,7,8}}
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,5,6},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,4,8,5,7] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,4,5,6,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,4,8,5,6] => {{1},{2},{3},{4,5,7},{6,8}}
=> {{1},{2},{3},{4,5,8},{6,7}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,3,4,7,8] => {{1},{2},{3,5},{4,6},{7},{8}}
=> {{1},{2},{3,6},{4,5},{7},{8}}
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,5,8,6,7] => {{1},{2,3},{4},{5},{6,7,8}}
=> {{1},{2,3},{4},{5},{6,7,8}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,8,6,7] => {{1},{2,3},{4,5},{6,7,8}}
=> {{1},{2,3},{4,5},{6,7,8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,4,5,7,8] => {{1},{2,3},{4,5,6},{7},{8}}
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,4,7,5,8] => {{1},{2,3},{4,5,6,7},{8}}
=> {{1},{2,3},{4,5,6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,7] => {{1},{2,3},{4,5,6,7,8}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,5,6,7},{8}}
=> {{1},{2,3},{4,5,6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,8,5,6] => {{1},{2,3},{4,5,7},{6,8}}
=> {{1},{2,3},{4,5,8},{6,7}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => {{1},{2,3},{4,5,6,7,8}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1},{2,3,6,7},{4,5},{8}}
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,5,8,2,4,6,7] => {{1},{2,3,5},{4,6,7,8}}
=> {{1},{2,3,6,7,8},{4,5}}
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7,8] => {{1},{2,3,4},{5},{6},{7},{8}}
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,4,2,3,5,7,6,8] => {{1},{2,3,4},{5},{6,7},{8}}
=> {{1},{2,3,4},{5},{6,7},{8}}
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,2,3,5,8,6,7] => {{1},{2,3,4},{5},{6,7,8}}
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,7,5,6,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,2,5,3,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6,8] => {{1},{2,3,4,5},{6,7},{8}}
=> {{1},{2,3,4,5},{6,7},{8}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,8,6,7] => {{1},{2,3,4,5},{6,7,8}}
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,4,2,6,3,5,7,8] => {{1},{2,3,4,5,6},{7},{8}}
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,2,7,3,8,5,6] => {{1},{2,3,4,5,7},{6,8}}
=> {{1},{2,3,4,5,8},{6,7}}
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,2,3,8] => {{1},{2,4,6},{3,5,7},{8}}
=> {{1},{2,7},{3,4,5,6},{8}}
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,5,2,3,4,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,2,3,4,7,6,8] => {{1},{2,3,4,5},{6,7},{8}}
=> {{1},{2,3,4,5},{6,7},{8}}
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,2,3,4,8,6,7] => {{1},{2,3,4,5},{6,7,8}}
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,6,3,4,7,8] => {{1},{2,3,5},{4,6},{7},{8}}
=> {{1},{2,3,6},{4,5},{7},{8}}
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1},{2,3,6,7},{4,5},{8}}
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,2,6,3,8,4,7] => {{1},{2,3,5},{4,6,7,8}}
=> {{1},{2,3,6,7,8},{4,5}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,7,3,4,6,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1},{2,3,6,7},{4,5},{8}}
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,8,4,6] => {{1},{2,3,5},{4,7},{6,8}}
=> {{1},{2,3,8},{4,5},{6,7}}
=> ? = 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000011
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0,0]
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0,0]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0,0]
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[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]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ?
=> ? = 2 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000248
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 82%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1,2}}
=> 2
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,2,4,5},{3}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,7,8}}
=> {{1,4,6,7,8},{2},{3},{5}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,5,6},{7},{8}}
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> {{1,2,6,7,8},{3},{4},{5}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,4,8,5,7] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,4,5,6,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> {{1,2,6,7,8},{3},{4},{5}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,4,8,5,6] => {{1},{2},{3},{4,5,7},{6,8}}
=> {{1,3,6,7,8},{2,4},{5}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,3,4,5,8,6] => {{1},{2},{3,4,5,6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,5,8,6,7] => {{1},{2,3},{4},{5},{6,7,8}}
=> {{1,4,5,6,8},{2},{3},{7}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,8,6,7] => {{1},{2,3},{4,5},{6,7,8}}
=> {{1,4,6,8},{2},{3},{5},{7}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,4,5,7,8] => {{1},{2,3},{4,5,6},{7},{8}}
=> {{1,2,3,6,8},{4},{5},{7}}
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,4,7,5,8] => {{1},{2,3},{4,5,6,7},{8}}
=> {{1,2,6,8},{3},{4},{5},{7}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,7] => {{1},{2,3},{4,5,6,7,8}}
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,5,6,7},{8}}
=> {{1,2,6,8},{3},{4},{5},{7}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,8,5,6] => {{1},{2,3},{4,5,7},{6,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => {{1},{2,3},{4,5,6,7,8}}
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1,2,5,8},{3},{4},{6},{7}}
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1,2,5,8},{3},{4,6},{7}}
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,5,8,2,4,6,7] => {{1},{2,3,5},{4,6,7,8}}
=> {{1,5,8},{2},{3},{4,6},{7}}
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7,8] => {{1},{2,3,4},{5},{6},{7},{8}}
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,4,2,3,5,7,6,8] => {{1},{2,3,4},{5},{6,7},{8}}
=> {{1,2,4,5,8},{3},{6},{7}}
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,2,3,5,8,6,7] => {{1},{2,3,4},{5},{6,7,8}}
=> {{1,4,5,8},{2},{3},{6},{7}}
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,7,5,6,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1,2,5,8},{3},{4},{6},{7}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,2,5,3,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6,8] => {{1},{2,3,4,5},{6,7},{8}}
=> {{1,2,4,8},{3},{5},{6},{7}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,8,6,7] => {{1},{2,3,4,5},{6,7,8}}
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,4,2,6,3,5,7,8] => {{1},{2,3,4,5,6},{7},{8}}
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,2,6,3,7,5,8] => {{1},{2,3,4,5,6,7},{8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7] => {{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,2,6,7,8,3,5] => {{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,4,2,7,3,5,6,8] => {{1},{2,3,4,5,6,7},{8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,2,7,3,8,5,6] => {{1},{2,3,4,5,7},{6,8}}
=> {{1,3,8},{2,4},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,8,3,5,6,7] => {{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,2,3,8] => {{1},{2,4,6},{3,5,7},{8}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,5,2,3,4,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,2,3,4,7,6,8] => {{1},{2,3,4,5},{6,7},{8}}
=> {{1,2,4,8},{3},{5},{6},{7}}
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,2,3,4,8,6,7] => {{1},{2,3,4,5},{6,7,8}}
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,6,3,4,7,8] => {{1},{2,3,5},{4,6},{7},{8}}
=> {{1,2,3,5,8},{4,6},{7}}
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1,2,5,8},{3},{4,6},{7}}
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,2,6,3,8,4,7] => {{1},{2,3,5},{4,6,7,8}}
=> {{1,5,8},{2},{3},{4,6},{7}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,7,3,4,6,8] => {{1},{2,3,5},{4,6,7},{8}}
=> {{1,2,5,8},{3},{4,6},{7}}
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,8,4,6] => {{1},{2,3,5},{4,7},{6,8}}
=> {{1,3,8},{2,5},{4,6},{7}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,8,3,4,6,7] => {{1},{2,3,5},{4,6,7,8}}
=> {{1,5,8},{2},{3},{4,6},{7}}
=> ? = 1
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 5 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ?
=> ?
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ?
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ?
=> ?
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ?
=> ?
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ?
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ?
=> ?
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ?
=> ?
=> ? = 4 + 2
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ?
=> ?
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> ?
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ?
=> ?
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ?
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000986
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 73%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> 2
[1,1,0,0]
=> [2] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,1,0,0,0]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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] => [8] => ([],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,7] => ([(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,6] => [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)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,2,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,1] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [1,1,2,2,2] => ([(1,7),(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)
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,1,1] => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,4,1] => [2,1,1,2,2] => ([(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)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5] => [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)
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,1] => [2,1,1,2,2] => ([(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)
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,5] => [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)
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => [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)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,3,1] => [2,1,2,1,2] => ([(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)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,6,1] => [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)
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7] => [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)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [1,1,3,1,2] => ([(1,6),(1,7),(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)
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,3,3,1] => [2,1,2,1,2] => ([(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)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [2,2,1,1,2] => ([(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)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [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)
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,5,1,1] => [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)
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,6,1] => [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)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,7] => [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)
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,7] => [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)
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,6,1] => [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)
=> ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,7] => [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)
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,7] => [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)
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,6,1] => [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)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [2,2,1,1,2] => ([(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)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [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)
=> ? = 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001126
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 73%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 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] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,2,1] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,1] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,4,1] => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,1] => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,5] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,3,1] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,6,1] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,3,3,1] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,6,1] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,6,1] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,6,1] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001691The number of kings in a graph. St000678The number of up steps after the last double rise of a Dyck path. St000895The number of ones on the main diagonal of an alternating sign matrix. St000675The number of centered multitunnels of a Dyck path. St000502The number of successions of a set partitions. St000234The number of global ascents of a permutation. St001479The number of bridges of a graph. St000025The number of initial rises of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000441The number of successions of a permutation. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000894The trace of an alternating sign matrix. St000717The number of ordinal summands of a poset. St000843The decomposition number of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St000456The monochromatic index of a connected graph. St000313The number of degree 2 vertices of a graph. St000884The number of isolated descents of a permutation. St001545The second Elser number of a connected graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. 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. St000061The number of nodes on the left branch of a binary tree. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000648The number of 2-excedences of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000732The number of double deficiencies of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001903The number of fixed points of a parking function.
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