Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000516
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000516: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [2,1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,3,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
Description
The number of stretching pairs of a permutation. This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Mp00128: Set partitions to compositionInteger compositions
Mp00038: Integer compositions reverseInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
St001438: Skew partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [[1],[]]
=> 0
{{1,2}}
=> [2] => [2] => [[2],[]]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [[1,1],[]]
=> 0
{{1,2,3}}
=> [3] => [3] => [[3],[]]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [[2,1],[]]
=> 0
{{1,3},{2}}
=> [2,1] => [1,2] => [[2,1],[]]
=> 0
{{1},{2,3}}
=> [1,2] => [2,1] => [[2,2],[1]]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [[1,1,1],[]]
=> 0
{{1,2,3,4}}
=> [4] => [4] => [[4],[]]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,2},{3,4}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [[2,1,1],[]]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,3},{2,4}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [[2,1,1],[]]
=> 0
{{1,4},{2,3}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1},{2,3,4}}
=> [1,3] => [3,1] => [[3,3],[2]]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [[2,2,1],[1]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [[2,1,1],[]]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [[2,2,1],[1]]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> 0
{{1,2,3,4,5}}
=> [5] => [5] => [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [[3,1,1],[]]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [[3,1,1],[]]
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [[3,1,1],[]]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [[3,2,2],[1,1]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [[3,1,1],[]]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [[3,1,1],[]]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [[3,2,2],[1,1]]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
Description
The number of missing boxes of a skew partition.
Matching statistic: St001727
Mp00174: Set partitions dual major index to intertwining numberSet partitions
Mp00164: Set partitions Chen Deng Du Stanley YanSet partitions
Mp00080: Set partitions to permutationPermutations
St001727: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> [1] => 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> [1,3,2] => 0
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2,3}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1},{2,3,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 2
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
{{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 2
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 2
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 0
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 0
{{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
{{1,3},{2,4,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => 2
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 0
{{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 0
{{1,4,5},{2,3}}
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 2
Description
The number of invisible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
St000609: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Mp00174: Set partitions dual major index to intertwining numberSet partitions
Mp00215: Set partitions Wachs-WhiteSet partitions
St000491: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2,3}}
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
{{1},{2,3,4}}
=> {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> 2
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> {{1,3,4},{2,5}}
=> 2
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> {{1,4},{2},{3,5}}
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
{{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> 2
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
{{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
{{1,4,5},{2,3}}
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4,5},{2,3}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Mp00174: Set partitions dual major index to intertwining numberSet partitions
Mp00217: Set partitions Wachs-White-rho Set partitions
St000497: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 0
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2,3}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
{{1},{2,3,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
{{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 2
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 0
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
{{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> 2
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 1
Description
The lcb statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000589
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000589: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> {{1}}
=> ? = 0
{{1,2}}
=> [2] => [1,1,0,0]
=> {{1,2}}
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000612
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000612: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> {{1}}
=> ? = 0
{{1,2}}
=> [2] => [1,1,0,0]
=> {{1,2}}
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Mp00220: Set partitions YipSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001857: Signed permutations ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 40%
Values
{{1}}
=> {{1}}
=> [1] => [1] => 0
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ? = 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ? = 0
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ? = 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => ? = 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ? = 0
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ? = 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ? = 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ? = 0
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => ? = 1
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ? = 2
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ? = 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ? = 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => ? = 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => ? = 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ? = 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => ? = 2
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => ? = 0
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => ? = 0
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
{{1,3,5},{2},{4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
{{1,3},{2},{4,5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => ? = 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ? = 0
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 1
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => ? = 2
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
{{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
{{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 3
{{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
{{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
{{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 3
{{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
Description
The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.