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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St001687
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001687: 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]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000143
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> []
=> []
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [3,2]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,2,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [3,2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [2,2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [3,3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,3,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> 0
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St001683
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [3,2,4,5,1,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [5,4,3,2,1,6,7] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [2,3,1,4,6,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [2,1,4,3,6,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [2,3,4,1,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [3,2,4,1,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [2,1,3,5,6,4,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [2,1,4,5,6,3,7] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [2,3,4,5,6,1,7] => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [3,2,4,5,6,1,7] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,4,3,5,6,1,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [3,4,2,5,6,1,7] => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [4,3,2,5,6,1,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [2,1,5,4,6,3,7] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [2,3,5,4,6,1,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [3,2,5,4,6,1,7] => ? = 1
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000371
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [2,4,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,3,4,5,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,5,4,3,1] => [3,2,4,5,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,5,4,2,1] => [2,4,3,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,5,4,2] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => [4,3,2,5,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => [3,5,4,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,2,1,5,3] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,1,5,2] => [5,3,4,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => [5,3,4,2,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,4,3,5,1] => [5,2,4,3,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [3,7,6,5,4,2,1] => [2,4,3,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [3,1,7,6,5,4,2] => [4,3,1,5,6,7,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => [2,3,1,5,6,7,4] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,3,1,7,6,5,4] => [3,2,1,5,6,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [4,7,6,5,3,2,1] => [2,3,5,4,6,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [4,1,7,6,5,3,2] => [3,5,4,1,6,7,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [4,2,7,6,5,3,1] => [3,5,4,2,6,7,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [4,2,1,7,6,5,3] => [2,5,4,1,6,7,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,4,2,7,6,5,3] => [1,5,4,2,6,7,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,3,7,6,5,2,1] => [2,5,4,3,6,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [4,3,1,7,6,5,2] => [5,3,4,1,6,7,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [4,3,2,7,6,5,1] => [5,3,4,2,6,7,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,3,2,1,7,6,5] => [2,3,4,1,6,7,5] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [2,4,3,7,6,5,1] => [5,2,4,3,6,7,1] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,1,7,6,5] => [3,2,4,1,6,7,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => [2,5,3,4,6,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [3,4,1,7,6,5,2] => [5,4,3,1,6,7,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [3,4,2,7,6,5,1] => [5,4,3,2,6,7,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [3,4,2,1,7,6,5] => [2,4,3,1,6,7,5] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,1,4,2,7,6,5] => [4,3,1,2,6,7,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [3,2,4,1,7,6,5] => [4,3,2,1,6,7,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,2,1,4,7,6,5] => [2,3,1,4,6,7,5] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,3,4,1,7,6,5] => [4,2,3,1,6,7,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [2,3,1,4,7,6,5] => [3,2,1,4,6,7,5] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [5,7,6,4,3,2,1] => [2,3,4,6,5,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [5,1,7,6,4,3,2] => [3,4,6,5,1,7,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => [3,4,6,5,2,7,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [5,2,1,7,6,4,3] => [2,4,6,5,1,7,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [5,3,7,6,4,2,1] => [2,4,6,5,3,7,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [5,3,1,7,6,4,2] => [4,3,6,5,1,7,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [5,3,2,7,6,4,1] => [4,3,6,5,2,7,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [5,3,2,1,7,6,4] => [2,3,6,5,1,7,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [2,5,3,7,6,4,1] => [4,2,6,5,3,7,1] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,5,3,1,7,6,4] => [3,2,6,5,1,7,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [2,1,5,3,7,6,4] => [2,1,6,5,3,7,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [5,4,7,6,3,2,1] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [5,4,1,7,6,3,2] => [3,6,4,5,1,7,2] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [5,4,2,7,6,3,1] => [3,6,4,5,2,7,1] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [5,4,2,1,7,6,3] => [2,6,4,5,1,7,3] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,5,4,2,7,6,3] => [1,6,4,5,2,7,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [5,4,3,7,6,2,1] => [2,6,4,5,3,7,1] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [5,4,3,1,7,6,2] => [6,3,4,5,1,7,2] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [5,4,3,2,7,6,1] => [6,3,4,5,2,7,1] => ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [2,5,4,3,7,6,1] => [6,2,4,5,3,7,1] => ? = 4
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,5,4,3,1,7,6] => [3,2,4,5,1,7,6] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ? = 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001685
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> [1] => 0
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => ? = 3
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000316
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> [1] => [1] => 0
[1,0,1,0]
=> [[.,.],.]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [.,[.,.]]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => 1
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 2
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,2,4,5,3] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,2,5,3,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,2,5,3,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,3,5,2,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,2,3,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => [1,2,3,4,5,7,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],[.,.]],.],.]
=> [5,1,2,3,4,6,7] => [1,2,3,4,6,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => [1,2,3,4,7,5,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],.],.],[[.,.],[.,.]]]
=> [7,5,6,1,2,3,4] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => [1,2,3,4,7,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> [6,5,7,1,2,3,4] => [1,2,3,4,5,7,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> [6,7,5,1,2,3,4] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [1,2,3,5,6,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => [1,2,3,5,6,4,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => [1,2,3,5,7,4,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => [1,2,3,6,7,4,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => [1,2,3,6,4,5,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => [1,2,3,4,5,7,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => [1,2,3,6,7,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => [1,2,3,6,4,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[[[.,.],.],[[.,[.,.]],.]],.]
=> [5,4,6,1,2,3,7] => [1,2,3,7,4,6,5] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => [1,2,3,4,6,7,5] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],.],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,1,2,3] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[[.,.],.]],.]]
=> [5,6,4,7,1,2,3] => [1,2,3,4,7,5,6] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],.]]]
=> [5,6,7,4,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,1,2,3] => [1,2,3,4,7,5,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[[.,.],.],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,1,2,3] => [1,2,3,4,5,7,6] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [1,2,4,5,6,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => [1,2,4,5,6,3,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => [1,2,4,5,7,3,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => [1,2,4,5,3,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => [1,2,4,5,3,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => [1,2,4,6,7,3,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => [1,2,4,6,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => [1,2,4,7,3,5,6] => ? = 3
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001682
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => [2,3,1] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => [1,2,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => [2,3,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => [2,1,3,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => [1,2,3,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,2,3] => [1,4,3,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [4,3,1,5,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => [4,2,3,1,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => [4,2,3,5,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => [4,2,1,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => [4,2,1,5,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,1,3,5] => [4,2,5,3,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => [4,1,2,3,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => [4,1,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,1,3,4] => [4,1,5,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [3,4,2,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => [3,4,2,5,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => [3,2,4,1,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => [3,2,4,5,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [3,2,1,4,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [3,2,1,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,5,4,2,1] => [3,1,2,4,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,1,2] => [3,1,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [3,5,1,2,4] => [3,1,5,4,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,1,2,4,5] => [3,5,4,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [6,5,4,2,3,1,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,5,4,2,3,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [2,3,4,6,7,5,1] => [6,5,4,2,1,3,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => [6,5,4,2,7,3,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,4,7,6,5,1] => [6,5,4,1,2,3,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [2,3,4,7,6,1,5] => [6,5,4,1,2,7,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [2,3,4,7,1,5,6] => [6,5,4,1,7,3,2] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => [6,5,3,4,2,1,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [2,3,5,4,6,1,7] => [6,5,3,4,2,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => [6,5,3,4,1,2,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [2,3,5,4,7,1,6] => [6,5,3,4,1,7,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,3,5,4,1,6,7] => [6,5,3,4,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [2,3,5,6,4,7,1] => [6,5,3,2,4,1,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [2,3,5,6,4,1,7] => [6,5,3,2,4,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [2,3,5,6,7,4,1] => [6,5,3,2,1,4,7] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [2,3,5,6,7,1,4] => [6,5,3,2,1,7,4] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [2,3,5,6,1,4,7] => [6,5,3,2,7,4,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [2,3,5,7,6,4,1] => [6,5,3,1,2,4,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [2,3,5,7,6,1,4] => [6,5,3,1,2,7,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [2,3,5,7,1,4,6] => [6,5,3,1,7,4,2] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [2,3,5,1,4,6,7] => [6,5,3,7,4,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => [6,5,2,3,4,1,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,3,6,5,4,1,7] => [6,5,2,3,4,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [2,3,6,5,7,4,1] => [6,5,2,3,1,4,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [2,3,6,5,7,1,4] => [6,5,2,3,1,7,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [2,3,6,5,1,4,7] => [6,5,2,3,7,4,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [2,3,6,7,5,4,1] => [6,5,2,1,3,4,7] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [2,3,6,7,5,1,4] => [6,5,2,1,3,7,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [2,3,6,7,1,4,5] => [6,5,2,1,7,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [2,3,6,1,4,5,7] => [6,5,2,7,4,3,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [2,3,7,6,5,4,1] => [6,5,1,2,3,4,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [2,3,7,6,5,1,4] => [6,5,1,2,3,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [2,3,7,6,1,4,5] => [6,5,1,2,7,4,3] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [2,3,7,1,4,5,6] => [6,5,1,7,4,3,2] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [2,3,1,4,5,6,7] => [6,5,7,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => [6,4,5,3,2,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,4,3,5,6,1,7] => [6,4,5,3,2,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,4,3,5,7,6,1] => [6,4,5,3,1,2,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,4,3,5,7,1,6] => [6,4,5,3,1,7,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [2,4,3,5,1,6,7] => [6,4,5,3,7,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => [6,4,5,2,3,1,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,3,6,5,1,7] => [6,4,5,2,3,7,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,3,6,7,5,1] => [6,4,5,2,1,3,7] => ? = 3
Description
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
Matching statistic: St001596
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 60%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [.,.]
=> [1,0]
=> [[1],[]]
=> 0
[1,0,1,0]
=> [.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[1,1,0,0]
=> [[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 0
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 0
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> 2
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 0
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 0
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 0
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 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],[]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 3
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2],[]]
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2],[]]
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ? = 4
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> ? = 3
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> ? = 3
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? = 4
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> ? = 4
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St000100
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The number of linear extensions of a poset.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 29%●distinct values known / distinct values provided: 20%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 29%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
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