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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000002
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000119
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000423
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000423: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000423: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Matching statistic: St000428
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
Matching statistic: St001411
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001411: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001411: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
Description
The number of patterns 321 or 3412 in a permutation.
A permutation is '''boolean''' if its principal order ideal in the (strong) Bruhat order is boolean.
It is shown in [1, Theorem 5.3] that a permutation is boolean if and only if it avoids the two patterns 321 and 3412.
Matching statistic: St000561
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 10
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,2,3}} in a set partition.
Matching statistic: St000436
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => ? = 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000437
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => ? = 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Matching statistic: St001604
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [[1],[]]
=> []
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4,5},{2},{3}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,4,5},{3}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,5},{2},{3,4}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2},{3,4,5}}
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3,4,5},{6}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,4,6},{5}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,4},{5,6}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,5},{4,6}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
{{1,2,4,5,6},{3}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,4,5},{3,6}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
{{1,2,4,6},{3,5}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2,4,6},{3},{5}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
{{1,2,5,6},{3,4}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0}
{{1,2,3}}
=> [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,4}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,10}
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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