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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St000834
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation w=[w1,...,wn] is either a position i such that wi−1<wi>wi+1 or n if wn>wn−1.
In other words, it is a peak in the word [w1,...,wn,0].
Matching statistic: St001280
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[]
=> []
=> [] => ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 11 => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 10 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1011 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 1101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1110 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1110 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1100 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1110 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1101 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1100 => 1
[]
=> []
=> [] => => ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ? => ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [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 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,1,1,2,1,1,1] => 1011110111 => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[]
=> []
=> [] => ? => ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,4] => 1000001000 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,5] => 1000010000 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [5,2,2,1] => 1000010101 => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2,2] => 1000101010 => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,8] => 1010000000 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [8,1,1] => 1000000011 => ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [7,1,1,1] => 1000000111 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1,1,1] => 1000001111 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 1
[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 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [6,4] => 1000001000 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [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 => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,4,1,1] => 1000100011 => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,3,2,2] => 1001001010 => ? = 4
[1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,1,1,1,1] => 1001011111 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [5,4,1] => 1000010001 => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4,2] => 1000100010 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,3] => 1000100100 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,3] => 1000100100 => ? = 3
[1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [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 => ? = 1
[1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => 1010111111 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ? => ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2,2] => 1000101010 => ? = 4
Description
The number of descents of a binary word.
Matching statistic: St000157
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [[1]]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ?
=> ? = 2
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ?
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,2,6,4,8,5,7] => ?
=> ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,1,7,8,9,5] => [[1,3,4,5,7,8,9],[2,6]]
=> ? = 2
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,1,8,9,10,6] => [[1,3,4,5,6,8,9,10],[2,7]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 2
[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,10,1,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[1,0,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,3,6,5,7,8,9] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,4,9,6,7,8] => [[1,2,3,4,6,7,8],[5,9]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,8,5,6,7,9] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,3,7,5,6,8,9] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8,9] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,3,7,5,6,9,8] => ?
=> ? = 3
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8,9] => ?
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5,9,7,8] => ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [6,1,2,3,4,5,9,7,8] => [[1,2,3,4,5,7,8],[6,9]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,6,10,8,9] => [[1,2,3,4,5,6,8,9],[7,10]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,1,2,3,4,5,10,7,8,9] => [[1,2,3,4,5,7,8,9],[6,10]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4,10,6,7,8,9] => [[1,2,3,4,6,7,8,9],[5,10]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[1,2,3,4,5,6,7,8,9,11],[10]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,8,7,9] => [[1,2,3,5,7,9],[4,6,8]]
=> ? = 3
[1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,9,8,10] => ?
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,9,1,2,4,5,6,7,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,8,9,1,7] => [[1,3,4,5,6,7,9],[2,8]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [6,1,2,9,3,4,5,7,8] => [[1,2,3,4,5,7,8],[6,9]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,9,4,6,7,8] => [[1,2,3,4,6,7,8],[5,9]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,10,2,3,4,5,6,7,9] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,10,3,4,5,6,8,9] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,10,4,5,7,8,9] => ?
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,11,1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [[1,2,3,4,5,6,9],[7,8]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [[1,2,3,4,5,6,7,10],[8,9]]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [[1,2,3,4,5,6,9],[7,8]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [[1,2,3,4,5,7,9],[6,8]]
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,9,3,4,6,8] => ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,8,5,6,7,9,10] => ?
=> ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,6,4,5,8,7,10,9] => ?
=> ? = 4
Description
The number of descents of a standard tableau.
Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Matching statistic: St000919
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 83%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => [.,.]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,1,2,3,4,5,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,1,2,3,4,8,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,1,2,3,6,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,1,2,3,6,5,8,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [4,1,2,5,6,3,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,1,8,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [5,1,2,8,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,1,2,3,8,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,1,2,3,8,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,3,7,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ?
=> ? = 2
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ?
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,8,3,5,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,2,6,4,8,5,7] => ?
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,6,4,5,8,7] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,7,4,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,1,7,8,9,5] => [[.,.],[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ? = 2
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,1,8,9,10,6] => [[.,.],[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,9,1,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 2
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000884
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 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]
=> [3,1,2,4] => 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]
=> [3,1,4,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,1,5,2,6,7,3] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [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]
=> [2,5,1,3,6,7,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,1,6,3,7,4] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,1,2,7,3,5,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,1,2,7,5,6] => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,1,6,7,3] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,1,3,7,5,6] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,7,4,6] => ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6] => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,1,6,7,4] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,1,7,3] => ? = 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,8,2,4,5,6,7] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,5,6,7] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,4,5,8,3,6,7] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3,8] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,2,6,7,8] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,6,7,2,8] => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,1,2,8,3,5,6,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 3
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [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]
=> [3,1,2,6,4,5,7,8] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,4,2,3,8,5,6,7] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,2,4,7,5,6,8] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,3,4,8,6,7] => ? = 2
Description
The number of isolated descents of a permutation.
A descent i is isolated if neither i+1 nor i−1 are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000035
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 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]
=> [3,1,2,4] => 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]
=> [3,1,4,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,1,5,2,6,7,3] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [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]
=> [2,5,1,3,6,7,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,1,6,3,7,4] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,1,2,7,3,5,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,1,2,7,5,6] => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,1,6,7,3] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,1,3,7,5,6] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,7,4,6] => ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6] => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,1,6,7,4] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,1,7,3] => ? = 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => ? = 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,8,2,4,5,6,7] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,5,6,7] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,4,5,8,3,6,7] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3,8] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,2,6,7,8] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,6,7,2,8] => ? = 2
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation w=[w1,...,wn] is either a position i such that wi−1<wi>wi+1 or 1 if w1>w2.
In other words, it is a peak in the word [0,w1,...,wn].
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000659
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [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]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000994
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 1
[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]
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => ? = 2
[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]
=> [2,4,1,5,6,7,3] => [3,1,7,2,4,5,6] => ? = 1
[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]
=> [3,4,1,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 2
[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]
=> [3,1,2,5,6,7,4] => [2,3,1,7,4,5,6] => ? = 2
[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]
=> [4,1,2,5,6,7,3] => [2,3,7,1,4,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,7,5] => [2,4,1,3,7,5,6] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [4,1,2,7,3,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => [4,7,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,5,2,6,7,4] => [2,4,1,7,3,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,5,2,7,6] => [2,5,1,3,4,7,6] => ? = 2
[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]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,6,2,7,5] => [2,5,1,3,7,4,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [5,1,7,2,3,4,6] => ? = 2
[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]
=> [4,5,1,6,2,7,3] => [3,5,7,1,2,4,6] => ? = 2
[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]
=> [2,1,3,6,4,7,5] => [2,1,3,5,7,4,6] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 3
[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]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[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]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,7] => [2,4,1,3,6,5,7] => ? = 2
[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]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,4,5,7,2,6] => [2,6,1,3,4,7,5] => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,4,6,7,2,5] => [2,6,1,3,7,4,5] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => [6,1,7,2,3,4,5] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => [3,6,7,1,2,4,5] => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => [3,4,6,7,1,2,5] => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5,7] => [2,5,1,3,6,4,7] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [5,1,6,7,2,3,4] => ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 3
[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]
=> [2,3,1,5,6,7,8,4] => [3,1,2,8,4,5,6,7] => ? = 2
[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]
=> [2,4,1,5,6,7,8,3] => [3,1,8,2,4,5,6,7] => ? = 1
[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]
=> [3,4,1,5,6,7,8,2] => [3,8,1,2,4,5,6,7] => ? = 2
[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]
=> [4,1,2,5,6,7,8,3] => [2,3,8,1,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [4,1,2,8,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => [4,8,1,2,3,5,6,7] => ? = 2
[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]
=> [5,1,2,3,6,7,8,4] => [2,3,4,8,1,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,4,5,2,7,8,6] => [2,5,1,3,4,8,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => [5,1,2,3,8,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [5,1,8,2,3,4,6,7] => ? = 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]
=> [4,5,1,6,2,7,8,3] => [3,5,8,1,2,4,6,7] => ? = 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation π is an index i such that π−1(i)<i>π(i). Analogously, a '''cycle valley''' is an index i such that π−1(i)>i<π(i).
Clearly, every cycle of π contains as many peaks as valleys.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000211The rank of the set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000703The number of deficiencies of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000386The number of factors DDU in a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000354The number of recoils of a permutation. St000201The number of leaf nodes in a binary tree. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St000702The number of weak deficiencies of a permutation. St000021The number of descents of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St000325The width of the tree associated to a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St000647The number of big descents of a permutation. St000353The number of inner valleys of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000710The number of big deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a 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. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000256The number of parts from which one can substract 2 and still get an integer partition. St001330The hat guessing number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001597The Frobenius rank of a skew partition. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St000028The number of stack-sorts needed to sort a permutation. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.
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