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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001868
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,5,2,4,3] => [1,5,2,4,3] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St000565
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000565: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000565: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [[1]]
=> {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
{{1},{2}}
=> [1,2] => [[1,2]]
=> {{1,2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
{{1,3},{2}}
=> [3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> [1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [[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]]
=> {{1,2,3,4,5}}
=> 0
Description
The major index of a set partition.
Let $\pi=B_1/B_2/\dots/B_k$ with $\min B_1<\min B_2<\dots<\min B_k$ a set partition. Let $d_i$ be the number of elements in $B_i$ larger than $\min B_{i+1}$. Then the major index of $\pi$ is $1d_1+2d_2+\dots+(k-1)d_{k-1}$.
Matching statistic: St001857
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
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}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{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}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => ? = 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ? = 0
{{1,2},{3,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ? = 1
{{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => ? = 2
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ? = 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => ? = 0
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => ? = 1
{{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ? = 0
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ? = 0
{{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ? = 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ? = 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ? = 0
{{1},{2},{3,4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ? = 0
{{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}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
{{1,2,3,5},{4}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,3},{4,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 1
{{1,2,3},{4},{5}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
{{1,2,4,5},{3}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
{{1,2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 0
{{1,2,4},{3},{5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
{{1,3,5},{2,4}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => ? = 1
{{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
{{1},{2,3,4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
{{1},{2,3,5},{4}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
{{1},{2,3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
{{1},{2,4,5},{3}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
{{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
{{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1},{2},{3,4,5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => ? = 0
{{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
{{1},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => ? = 0
{{1},{2},{3},{4,5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ? = 0
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
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.
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