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Your data matches 105 different statistics following compositions of up to 3 maps.
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Matching statistic: St000623
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(load all 7 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000623: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 0
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 0
Description
The number of occurrences of the pattern 52341 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to have a complete parabolic bundle structure [1].
Matching statistic: St001175
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> [1,1]
=> [1]
=> 0
{{1},{2}}
=> [1,1]
=> [2]
=> []
=> ? = 0
{{1,2,3}}
=> [3]
=> [3]
=> []
=> ? = 0
{{1,2},{3}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [3,1]
=> [1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [3,1]
=> [1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [3,1]
=> [1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [5]
=> []
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,3,4},{2,5,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,3,5},{2,4,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,3,6},{2,4,5}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,4,5},{2,3,6}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,4,6},{2,3,5}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
{{1,5,6},{2,3,4}}
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,4}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000022
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The number of fixed points of a permutation.
Matching statistic: St000153
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St001465
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000031
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 92%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 0 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4} + 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St001498
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? = 0
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000221
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,1}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000279
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000279: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000279: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,1}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations.
Matching statistic: St000375
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 91%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,1}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,4}
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001890The maximum magnitude of the Möbius function of a poset. St001866The nesting alignments of a signed permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001429The number of negative entries in a signed permutation. St000068The number of minimal elements in a poset. St001862The number of crossings of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001768The number of reduced words of a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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