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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St001032
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
St001032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 3
Description
The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path.
In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1.
The generating function is given in [1].
Matching statistic: St000148
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
Description
The number of odd parts of a partition.
Matching statistic: St000475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001247
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
Description
Sum of the odd parts of a partition.
Matching statistic: St000247
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [[1,2]]
=> {{1,2}}
=> 2
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 3
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => 11 => 11 => 2
[1,1,0,0]
=> [2] => 10 => 00 => 0
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 100 => 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 100 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 010 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0011 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1001 => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1001 => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0101 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1100 => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0000 => 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1100 => 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1100 => 2
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 0000 => 0
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1010 => 2
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1010 => 2
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1010 => 2
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0110 => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00111 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10011 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 10011 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 01011 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11001 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 11001 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 11001 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 00001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10101 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 10101 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 10101 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 01101 => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11100 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 00100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10000 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 10000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11100 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 00100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 11100 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 11100 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 00100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 10000 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 01000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11010 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000392
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,1]
=> 11 => 2
[1,1,0,0]
=> [2,1] => [2]
=> 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 111 => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 01 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 01 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 01 => 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 011 => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 011 => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 011 => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 011 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 011 => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 00 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 011 => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 011 => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 011 => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 011 => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 00 => 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 011 => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 011 => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 0111 => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 0111 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 0111 => 3
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000445
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
Description
The number of rises of length 1 of a Dyck path.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000878The number of ones minus the number of zeros of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000160The multiplicity of the smallest part of a partition. St000696The number of cycles in the breakpoint graph of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000819The propagating number of a perfect matching. St000022The number of fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000389The number of runs of ones of odd length in a binary word. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000931The number of occurrences of the pattern UUU in a Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000221The number of strong fixed points of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000359The number of occurrences of the pattern 23-1. St000237The number of small exceedances. St001524The degree of symmetry of a binary word. St001948The number of augmented double ascents of a permutation. St000836The number of descents of distance 2 of a permutation. St000239The number of small weak excedances. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
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