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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000222
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Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3
Description
The number of alignments in the permutation.
Matching statistic: St000743
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000743: Standard tableaux ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000743: Standard tableaux ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> []
=> 0
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,0,0]
=> []
=> []
=> []
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [[1,2,3,10],[4,5,6],[7,8],[9]]
=> ? ∊ {2,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,5],[3,4,9],[6,7],[8]]
=> ? ∊ {2,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,9],[4,5],[6,7],[8]]
=> ? ∊ {2,3,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,2,8],[3,4],[5,6],[7]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3],[2,5],[4,7],[6]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,9],[4,5,6],[7],[8]]
=> ? ∊ {2,3,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,2,5],[3,4,8],[6],[7]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,2,3,8],[4,5],[6],[7]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,7],[3,4],[5],[6]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3],[2,6],[4],[5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,9],[4,5,6],[7,8]]
=> ? ∊ {2,3,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,2,3,8],[4,5],[6,7]]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,7],[3,4],[5,6]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,2,3,8],[4,5,6],[7]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,5],[3,4,7],[6]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,7],[4,5],[6]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,6],[4],[5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,2,3,7],[4,5,6]]
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,2,3,6],[4,5]]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,2,3,4,15],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,7],[4,5,6,14],[8,9,10],[11,12],[13]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,4,14],[5,6,7],[8,9,10],[11,12],[13]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [[1,2,5],[3,4,8],[6,7,12],[9,10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10],[11,12],[13]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,7],[4,5,6,13],[8,9],[10,11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,13],[5,6,7],[8,9],[10,11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [[1,2,3,12],[4,5,6],[7,8],[9,10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [[1,2,5],[3,4,11],[6,7],[8,9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,2,3,4,12],[5,6],[7,8],[9,10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,2,3,11],[4,5],[6,7],[8,9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [[1,2,10],[3,4],[5,6],[7,8],[9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10,11],[12],[13]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,7],[4,5,6,13],[8,9,10],[11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,4,13],[5,6,7],[8,9,10],[11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [[1,2,3,12],[4,5,6],[7,8,9],[10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [[1,2,5],[3,4,8],[6,7,11],[9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [[1,2,3,4,13],[5,6,7,8],[9,10],[11],[12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [[1,2,3,7],[4,5,6,12],[8,9],[10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [[1,2,3,4,12],[5,6,7],[8,9],[10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [[1,2,3,11],[4,5,6],[7,8],[9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [[1,2,5],[3,4,10],[6,7],[8],[9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [[1,2,3,4,11],[5,6],[7,8],[9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [[1,2,3,10],[4,5],[6,7],[8],[9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,9],[3,4],[5,6],[7],[8]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [[1,2,3,4,12],[5,6,7,8],[9],[10],[11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [[1,2,3,7],[4,5,6,11],[8],[9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [[1,2,3,4,11],[5,6,7],[8],[9],[10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [[1,2,3,10],[4,5,6],[7],[8],[9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,2,5],[3,4,9],[6],[7],[8]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [[1,2,3,4,10],[5,6],[7],[8],[9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,9],[4,5],[6],[7],[8]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,9],[5],[6],[7],[8]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10,11],[12,13]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,7],[4,5,6,13],[8,9,10],[11,12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,4,13],[5,6,7],[8,9,10],[11,12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [[1,2,3,12],[4,5,6],[7,8,9],[10,11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [[1,2,3,4,13],[5,6,7,8],[9,10],[11,12]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [[1,2,3,7],[4,5,6,12],[8,9],[10,11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,12],[5,6,7],[8,9],[10,11]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [[1,2,3,11],[4,5,6],[7,8],[9,10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [[1,2,5],[3,4,10],[6,7],[8,9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [[1,2,3,4,11],[5,6],[7,8],[9,10]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [[1,2,3,10],[4,5],[6,7],[8,9]]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The number of entries in a standard Young tableau such that the next integer is a neighbour.
Matching statistic: St000770
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 86%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 4
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 6
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [[5,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [[5,5],[4]]
=> [4]
=> 4
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000937
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 71%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 5
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 5
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [[5,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [[5,5],[4]]
=> [4]
=> 5
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000259
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000777
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {0,0,1} + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1} + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1} + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2} + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5 = 4 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 71%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0
[1,0,1,0]
=> [2,1] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [1,2] => [1] => ([],1)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => ([],2)
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => ([],2)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000264
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 29%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,5,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [4,1,2,5,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000460
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 2
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000474
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 43%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 4
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 4
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 4
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001175The size of a partition minus the hook length of the base cell. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001571The Cartan determinant of the integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000632The jump number of the poset. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001822The number of alignments of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000876The number of factors in the Catalan decomposition of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000628The balance of a binary word. St000307The number of rowmotion orbits of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000891The number of distinct diagonal sums of a permutation matrix.
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