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Your data matches 345 different statistics following compositions of up to 3 maps.
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Matching statistic: St000481
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,0,1,0]
=> [1,2] => [2,1] => [2]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [3]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [4]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,4,3] => [2,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => [2,2]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,3,1] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [3,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,3,4,2] => [3,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,1,5,4] => [3,2]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => [5]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => [3,2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,1,5,3] => [5]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,5,3,4,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,5,3,1,4] => [4,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,5,1,4,3] => [4,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,4,5,3] => [3,2]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [3,2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [5]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,4,1,5,2] => [3,2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,5,2,4,1] => [4,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => [5]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,5,1,4,2] => [2,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5]
=> 0
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001092
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> {{1}}
=> [1]
=> 0
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> [1,1]
=> 0
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> [2]
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> 0
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> [3]
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1]
=> 0
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1]
=> 0
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2]
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> 0
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2]
=> 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2]
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,5},{3}}
=> [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2]
=> 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001174
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [2,1,3,4,6,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,3,6,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [2,1,4,3,6,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,3,6,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,6,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001273
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001273: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001273: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The projective dimension of the first term in an injective coresolution of the regular module.
The algebra has the double centraliser property when 0 is returned and it is 1-Gorenstein in case a number < =1 is returned.
Matching statistic: St001803
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [[1,2,5],[3],[4]]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [[1,2,3],[4],[5]]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [[1,2,4],[3],[5]]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [[1,2,6],[3],[4],[5]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [[1,2,5],[3],[4],[6]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [[1,2,5],[3],[4],[6]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [[1,2,5,6],[3],[4]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [[1,2,4],[3],[5],[6]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [[1,2,4,6],[3],[5]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [[1,2,4],[3],[5],[6]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [[1,2,3,5],[4],[6]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [[1,2,4],[3,6],[5]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [[1,2,4,5],[3],[6]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [[1,2,3],[4],[5],[6]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [[1,2,3,6],[4],[5]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [[1,2,3,5],[4],[6]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [[1,2,3,5],[4],[6]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [[1,2,3,5,6],[4]]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [[1,2,3],[4],[5],[6]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [[1,2,3],[4,6],[5]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [[1,2,4],[3],[5],[6]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [[1,2,4],[3,6],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [[1,2,4],[3,6],[5]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [[1,2,3],[4,5],[6]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [[1,2,4,5],[3],[6]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000007
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 2 = 1 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000314
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,3,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [4,3,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [4,1,3,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,4,2,5,3] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,5,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,4,3,2] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [5,4,1,3,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [4,3,1,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [5,1,4,3,2] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,2,5,3] => [4,1,5,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [5,4,3,1,2] => 1 = 0 + 1
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000654
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => [4,2,3,1] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,4,3,1] => [2,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,3,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => [4,5,2,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,4,2,5,1] => [3,5,2,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => [3,5,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => [5,3,4,1,2] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [2,4,5,3,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [2,3,5,4,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,4,3,1] => [5,2,4,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => [4,5,1,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,5,4,1,2] => [3,5,4,1,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,4,1,5,3] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,6,4,5,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,5,6,3,4,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,4,5,3,6,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,6,3,4,1,5] => [2,6,5,4,1,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,5,6,2,3,1] => [4,6,5,2,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [3,4,2,5,6,1] => [3,6,2,5,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,4,5,2,6,1] => [3,6,5,2,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,5,1] => [6,3,5,2,4,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,6,2,3,1,4] => [5,6,2,4,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,5,2,3,1] => [4,6,5,2,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,5,2,4,6,1] => [3,6,2,5,4,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,3,1,4,5] => [6,2,5,1,4,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,4,5,6,1,2] => [3,6,5,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,4,5,1,2] => [3,6,5,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,5,3,6,1,2] => [4,6,3,5,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,4,1,2,5] => [6,3,5,1,4,2] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [2,4,5,6,3,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,6,4,5,3,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [2,3,5,6,4,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,3,4,6,1,5] => [2,6,5,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [2,6,3,5,4,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,6,2,4,3,1] => [5,6,2,4,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,5,2,6,3,1] => [4,6,2,5,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,2,6,4,1] => [3,6,2,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,4,3,1,5] => [6,2,5,4,1,3] => 1 = 0 + 1
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000701
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,6,4,5,1] => [[.,.],[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,5,6,3,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,4,5,3,6,1] => [[.,.],[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,6,3,4,1,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,5,6,2,3,1] => [[[.,.],[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [3,4,2,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,4,5,2,6,1] => [[[.,.],.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,5,1] => [[[[.,.],.],[.,[.,.]]],.]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,6,2,3,1,4] => [[[.,.],[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,5,2,3,1] => [[[.,.],[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,5,2,4,6,1] => [[[.,.],.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,3,1,4,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,4,5,6,1,2] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,4,5,1,2] => [[.,[.,.]],[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,5,3,6,1,2] => [[[.,[.,.]],.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,4,1,2,5] => [[[.,[.,.]],[.,[.,.]]],.]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [2,4,5,6,3,1] => [[.,.],[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,6,4,5,3,1] => [[.,.],[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [2,3,5,6,4,1] => [[.,.],[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,3,4,6,1,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [2,6,3,5,4,1] => [[.,.],[[.,[[.,.],.]],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,6,2,4,3,1] => [[[.,.],[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,5,2,6,3,1] => [[[.,.],[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,2,6,4,1] => [[[.,.],.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,4,3,1,5] => [[[.,.],[[.,.],[.,.]]],.]
=> 1 = 0 + 1
Description
The protection number of a binary tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000920
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
The following 335 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000451The length of the longest pattern of the form k 1 2. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. 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. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000659The number of rises of length at least 2 of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St000929The constant term of the character polynomial of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001737The number of descents of type 2 in a permutation. St001933The largest multiplicity of a part in an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001128The exponens consonantiae of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000455The second largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001728The number of invisible descents of a permutation. St000741The Colin de Verdière graph invariant. St001520The number of strict 3-descents. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000374The number of exclusive right-to-left minima of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000768The number of peaks in an integer composition. St000914The sum of the values of the Möbius function of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000871The number of very big ascents of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000259The diameter of a connected graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001960The number of descents of a permutation minus one if its first entry is not one. St000764The number of strong records in an integer composition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000635The number of strictly order preserving maps of a poset into itself. St000658The number of rises of length 2 of a Dyck path. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001139The number of occurrences of hills of size 2 in a Dyck path. St001964The interval resolution global dimension of a poset. St001868The number of alignments of type NE of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000862The number of parts of the shifted shape of a permutation. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St001651The Frankl number of a lattice. St000670The reversal length of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000630The length of the shortest palindromic decomposition of a binary word. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000534The number of 2-rises of a permutation. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001864The number of excedances of a signed permutation. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000669The number of permutations obtained by switching ascents or descents of size 2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000441The number of successions of a permutation. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000928The sum of the coefficients of the character polynomial of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000031The number of cycles in the cycle decomposition of a permutation. St001052The length of the exterior of a permutation. St000649The number of 3-excedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000552The number of cut vertices of a graph. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000963The 2-shifted major index of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000056The decomposition (or block) number of a permutation. St000090The variation of a composition. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000761The number of ascents in an integer composition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001151The number of blocks with odd minimum. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001569The maximal modular displacement of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001673The degree of asymmetry of an integer composition. St001735The number of permutations with the same set of runs. St001896The number of right descents of a signed permutations. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000903The number of different parts of an integer composition. St001267The length of the Lyndon factorization of the binary word. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001061The number of indices that are both descents and recoils of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001638The book thickness of a graph. St001644The dimension of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.
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