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Your data matches 158 different statistics following compositions of up to 3 maps.
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Matching statistic: St000486
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [3,1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,3,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,3] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => [4,1,3,2] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,2,4] => [3,1,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => [3,1,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,3] => [4,1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,1,4] => [2,4,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,4,3,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,4] => [5,1,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,2,5,3,4] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,3] => [4,1,5,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,4,2,3] => [5,1,4,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,2,5] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => [3,1,5,4,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,3] => [2,5,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,2,4] => [5,3,1,4,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => [5,1,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,4,1,2,3] => [5,4,1,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,1,5,2,3] => [4,1,5,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,3,1,5,2] => [4,3,1,5,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,1,4,5] => [2,5,1,4,3] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000618
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[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],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2}
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000620
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1}
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [4,2]
=> [2]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [4,2]
=> [2]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [4,2]
=> [2]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [4,2]
=> [2]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [3,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000402
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1 = 0 + 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,3,2,5,4,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,8,7,6,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [6,3,2,10,9,1,8,7,5,4] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [6,3,2,10,9,1,8,7,5,4] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [8,3,2,10,9,7,6,1,5,4] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [10,3,2,9,8,7,6,5,4,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [10,3,2,9,8,7,6,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [8,3,2,10,9,7,6,1,5,4] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [10,3,2,9,8,7,6,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [10,3,2,9,8,7,6,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [10,3,2,9,8,7,6,5,4,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [8,5,4,3,2,10,9,1,7,6] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [10,5,4,3,2,9,8,7,6,1] => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [10,5,4,3,2,9,8,7,6,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [8,7,4,3,10,9,2,1,6,5] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [10,7,4,3,9,8,2,6,5,1] => ? ∊ {0,0,0,0,0,1,1} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [10,9,4,3,8,7,6,5,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [6,3,2,5,4,1,8,7,10,9,12,11] => [6,3,2,12,11,1,10,9,8,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [6,3,2,5,4,1,8,7,12,11,10,9] => [6,3,2,12,11,1,10,9,8,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [6,3,2,5,4,1,10,9,8,7,12,11] => [6,3,2,12,11,1,10,9,8,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [6,3,2,5,4,1,12,9,8,11,10,7] => [6,3,2,12,11,1,10,9,8,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [6,3,2,5,4,1,12,11,10,9,8,7] => [6,3,2,12,11,1,10,9,8,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [8,3,2,5,4,7,6,1,10,9,12,11] => [8,3,2,12,11,10,9,1,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [8,3,2,5,4,7,6,1,12,11,10,9] => [8,3,2,12,11,10,9,1,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [10,3,2,5,4,7,6,9,8,1,12,11] => [10,3,2,12,11,9,8,7,6,1,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [10,3,2,5,4,9,8,7,6,1,12,11] => [10,3,2,12,11,9,8,7,6,1,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [8,3,2,7,6,5,4,1,10,9,12,11] => [8,3,2,12,11,10,9,1,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [8,3,2,7,6,5,4,1,12,11,10,9] => [8,3,2,12,11,10,9,1,7,6,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [10,3,2,7,6,5,4,9,8,1,12,11] => [10,3,2,12,11,9,8,7,6,1,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> [10,3,2,9,6,5,8,7,4,1,12,11] => [10,3,2,12,11,9,8,7,6,1,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [10,3,2,9,8,7,6,5,4,1,12,11] => [10,3,2,12,11,9,8,7,6,1,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]
=> [8,5,4,3,2,7,6,1,10,9,12,11] => [8,5,4,3,2,12,11,1,10,9,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
=> [8,5,4,3,2,7,6,1,12,11,10,9] => [8,5,4,3,2,12,11,1,10,9,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> [10,5,4,3,2,7,6,9,8,1,12,11] => [10,5,4,3,2,12,11,9,8,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> [10,5,4,3,2,9,8,7,6,1,12,11] => [10,5,4,3,2,12,11,9,8,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]
=> [8,7,4,3,6,5,2,1,10,9,12,11] => [8,7,4,3,12,11,2,1,10,9,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]
=> [8,7,4,3,6,5,2,1,12,11,10,9] => [8,7,4,3,12,11,2,1,10,9,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]
=> [10,7,4,3,6,5,2,9,8,1,12,11] => [10,7,4,3,12,11,2,9,8,1,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> [12,7,4,3,6,5,2,9,8,11,10,1] => [12,7,4,3,11,10,2,9,8,6,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]
=> [12,7,4,3,6,5,2,11,10,9,8,1] => [12,7,4,3,11,10,2,9,8,6,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [10,9,4,3,6,5,8,7,2,1,12,11] => [10,9,4,3,12,11,8,7,2,1,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> [12,9,4,3,6,5,8,7,2,11,10,1] => [12,9,4,3,11,10,8,7,2,6,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [10,9,4,3,8,7,6,5,2,1,12,11] => [10,9,4,3,12,11,8,7,2,1,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]
=> [12,9,4,3,8,7,6,5,2,11,10,1] => [12,9,4,3,11,10,8,7,2,6,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [10,7,6,5,4,3,2,9,8,1,12,11] => [10,7,6,5,4,3,2,12,11,1,9,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [10,9,6,5,4,3,8,7,2,1,12,11] => [10,9,6,5,4,3,12,11,2,1,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,6),(4,5),(7,8),(10,11)]
=> [12,9,6,5,4,3,8,7,2,11,10,1] => [12,9,6,5,4,3,11,10,2,8,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [10,9,8,5,4,7,6,3,2,1,12,11] => [10,9,8,5,4,12,11,3,2,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
=> [12,9,8,5,4,7,6,3,2,11,10,1] => [12,9,8,5,4,11,10,3,2,7,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [(1,12),(2,11),(3,8),(4,5),(6,7),(9,10)]
=> [12,11,8,5,4,7,6,3,10,9,2,1] => [12,11,8,5,4,10,9,3,7,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
Description
Half the size of the symmetry class of a permutation.
The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]).
This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.
Matching statistic: St000659
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[1,1,0,0]
=> []
=> []
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1}
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {0,0,0}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [5,2,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [9,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [6,6,1]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [9,4,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [8,4,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [9,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [9,5]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [8,5]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [9,4]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [9,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000781
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St000003
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> []
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2}
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000212
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> []
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
Description
The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.
Summing over all partitions of $n$ yields the sequence
$$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$
which is [[oeis:A237770]].
The references in this sequence of the OEIS indicate a connection with Baxter permutations.
Matching statistic: St000278
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> []
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
The following 148 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000326The position of the first one in a binary word after appending a 1 at the end. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001568The smallest positive integer that does not appear twice in the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St000897The number of different multiplicities of parts of an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001737The number of descents of type 2 in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001432The order dimension of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000284The Plancherel distribution on integer partitions. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001587Half of the largest even part of an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000741The Colin de Verdière graph invariant. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001820The size of the image of the pop stack sorting operator. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St001638The book thickness of a graph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000628The balance of a binary word. St000920The logarithmic height of a Dyck path. St001569The maximal modular displacement of a permutation. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000630The length of the shortest palindromic decomposition of a binary word. St001884The number of borders of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset.
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