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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000463
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{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] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The number of admissible inversions of a permutation.
Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$.
An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions:
$1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{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] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,4,6,7,5] => ? = 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,4,7,5,6] => ? = 2
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,4,7,5,6] => ? = 2
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,3,5,6,7,4] => ? = 3
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,3,5,6,4,7] => ? = 2
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,3,5,6,4,7] => ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,3,5,7,4,6] => ? = 3
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,3,5,4,6,7] => ? = 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,3,5,4,6,7] => ? = 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,3,5,7,4,6] => ? = 3
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => ? = 2
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,3,5,4,6,7] => ? = 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,3,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => ? = 4
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,2,3,6,4,5,7] => ? = 2
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,2,3,6,4,5,7] => ? = 2
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,3,7,4,5,6] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,3,7,4,5,6] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,2,3,6,4,5,7] => ? = 2
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,2,3,6,4,5,7] => ? = 2
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,3,7,4,6,5] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,3,4,6,7,5] => ? = 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,3,7,4,5,6] => ? = 3
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,3,4,7,5,6] => ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,3,7,4,5,6] => ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,3,4,7,5,6] => ? = 2
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [1,2,4,5,6,7,3] => ? = 4
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [1,2,4,5,6,3,7] => ? = 3
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [1,2,4,5,6,3,7] => ? = 3
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,2,4,5,7,3,6] => ? = 4
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [1,2,4,5,3,6,7] => ? = 2
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [1,2,4,5,3,6,7] => ? = 2
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [1,2,4,5,7,3,6] => ? = 4
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [1,2,4,5,3,7,6] => ? = 3
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [1,2,4,5,3,6,7] => ? = 2
{{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [1,2,4,5,3,6,7] => ? = 2
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,2,4,6,7,3,5] => ? = 5
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [1,2,4,6,3,5,7] => ? = 3
{{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => [1,2,4,6,3,5,7] => ? = 3
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,2,4,7,3,5,6] => ? = 4
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [1,2,4,3,5,6,7] => ? = 1
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [1,2,4,3,5,6,7] => ? = 1
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,2,4,7,3,5,6] => ? = 4
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [1,2,4,3,5,7,6] => ? = 2
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [1,2,4,3,5,6,7] => ? = 1
{{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => [1,2,4,3,5,6,7] => ? = 1
{{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [1,2,4,6,7,3,5] => ? = 5
{{1,2,4,6},{3,7},{5}}
=> [2,4,7,6,5,1,3] => [1,2,4,6,3,7,5] => ? = 4
{{1,2,4,6},{3},{5,7}}
=> [2,4,3,6,7,1,5] => [1,2,4,6,3,5,7] => ? = 3
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St001841
Mp00112: Set partitions —complement⟶ Set partitions
St001841: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
St001841: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
Values
{{1}}
=> {{1}}
=> 0
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 2
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 3
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 2
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 4
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,7},{6,8}}
=> ? = 1
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? = 2
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,2},{3,5},{4,8},{6,7}}
=> ? = 3
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3},{2,8},{4,5},{6,7}}
=> ? = 5
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? = 6
{{1,8},{2,4},{3,5},{6,7}}
=> {{1,8},{2,3},{4,6},{5,7}}
=> ? = 7
{{1,7},{2,4},{3,5},{6,8}}
=> {{1,3},{2,8},{4,6},{5,7}}
=> ? = 6
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,2},{3,8},{4,6},{5,7}}
=> ? = 5
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,2},{3,6},{4,8},{5,7}}
=> ? = 4
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,2},{3,6},{4,7},{5,8}}
=> ? = 3
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,2},{3,5},{4,7},{6,8}}
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2},{3,5},{4,6},{7,8}}
=> ? = 1
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2},{3,6},{4,5},{7,8}}
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,2},{3,7},{4,5},{6,8}}
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,2},{3,7},{4,6},{5,8}}
=> ? = 4
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,2},{3,7},{4,8},{5,6}}
=> ? = 5
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? = 6
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,3},{2,8},{4,7},{5,6}}
=> ? = 7
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,8},{2,3},{4,7},{5,6}}
=> ? = 8
{{1,8},{2,6},{3,4},{5,7}}
=> {{1,8},{2,4},{3,7},{5,6}}
=> ? = 9
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2,8},{3,7},{5,6}}
=> ? = 8
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2,7},{3,8},{5,6}}
=> ? = 7
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,3},{2,7},{4,8},{5,6}}
=> ? = 6
{{1,4},{2,7},{3,5},{6,8}}
=> {{1,3},{2,7},{4,6},{5,8}}
=> ? = 5
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,3},{2,7},{4,5},{6,8}}
=> ? = 4
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,3},{2,6},{4,5},{7,8}}
=> ? = 3
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,6},{2,3},{4,5},{7,8}}
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,7},{2,3},{4,5},{6,8}}
=> ? = 5
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,3},{4,6},{5,8}}
=> ? = 6
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,7},{2,3},{4,8},{5,6}}
=> ? = 7
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,7},{2,4},{3,8},{5,6}}
=> ? = 8
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,7},{2,8},{3,4},{5,6}}
=> ? = 9
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,8},{2,7},{3,4},{5,6}}
=> ? = 10
{{1,8},{2,7},{3,5},{4,6}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> ? = 11
{{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> ? = 10
{{1,6},{2,8},{3,5},{4,7}}
=> {{1,7},{2,5},{3,8},{4,6}}
=> ? = 9
{{1,5},{2,8},{3,6},{4,7}}
=> {{1,7},{2,5},{3,6},{4,8}}
=> ? = 8
{{1,4},{2,8},{3,6},{5,7}}
=> {{1,7},{2,4},{3,6},{5,8}}
=> ? = 7
{{1,3},{2,8},{4,6},{5,7}}
=> {{1,7},{2,4},{3,5},{6,8}}
=> ? = 6
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,6},{2,4},{3,5},{7,8}}
=> ? = 5
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,4},{2,6},{3,5},{7,8}}
=> ? = 4
{{1,3},{2,7},{4,6},{5,8}}
=> {{1,4},{2,7},{3,5},{6,8}}
=> ? = 5
{{1,4},{2,7},{3,6},{5,8}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> ? = 6
{{1,5},{2,7},{3,6},{4,8}}
=> {{1,5},{2,7},{3,6},{4,8}}
=> ? = 7
{{1,6},{2,7},{3,5},{4,8}}
=> {{1,5},{2,7},{3,8},{4,6}}
=> ? = 8
{{1,7},{2,6},{3,5},{4,8}}
=> {{1,5},{2,8},{3,7},{4,6}}
=> ? = 9
{{1,8},{2,6},{3,5},{4,7}}
=> {{1,8},{2,5},{3,7},{4,6}}
=> ? = 10
{{1,8},{2,5},{3,6},{4,7}}
=> {{1,8},{2,5},{3,6},{4,7}}
=> ? = 9
Description
The number of inversions of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
A pair $(i,j)$ is an inversion of the word $w$ if $w_i > w_j$.
Matching statistic: St001341
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001341: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001341: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => ([(5,7),(6,7)],8)
=> ? = 2
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 3
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 5
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 3
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => ([(3,4),(5,7),(6,7)],8)
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 5
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 6
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => ([(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 5
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => ([(2,3),(4,7),(5,7),(6,7)],8)
=> ? = 4
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 3
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => ([(1,2),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 5
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => ([(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 6
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => ([(1,7),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => ([(1,7),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 8
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 9
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 11
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 10
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => ([(1,7),(2,6),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 9
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => ([(1,7),(2,5),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 8
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => ([(1,6),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7)],8)
=> ? = 7
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => ([(1,2),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => ([(2,6),(3,4),(3,7),(4,7),(5,6),(5,7)],8)
=> ? = 6
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => ([(2,7),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 8
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9
Description
The number of edges in the center of a graph.
The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001579
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001579: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001579: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
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] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 5
{{1,3,4,5,6},{2,7}}
=> [3,7,4,5,6,1,2] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 4
{{1,3,4,5,6},{2},{7}}
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 4
{{1,3,4,5,7},{2,6}}
=> [3,6,4,5,7,2,1] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 5
{{1,3,4,5},{2,6,7}}
=> [3,6,4,5,1,7,2] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5},{2,6},{7}}
=> [3,6,4,5,1,2,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5,7},{2},{6}}
=> [3,2,4,5,7,6,1] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 5
{{1,3,4,5},{2,7},{6}}
=> [3,7,4,5,1,6,2] => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 4
{{1,3,4,5},{2},{6,7}}
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5},{2},{6},{7}}
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,6,7},{2,5}}
=> [3,5,4,6,2,7,1] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 6
{{1,3,4,6},{2,5,7}}
=> [3,5,4,6,7,1,2] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,6},{2,5},{7}}
=> [3,5,4,6,2,1,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,7},{2,5,6}}
=> [3,5,4,7,6,2,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,7},{2,5},{6}}
=> [3,5,4,7,2,6,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,6,7},{2},{5}}
=> [3,2,4,6,5,7,1] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 6
{{1,3,4,6},{2,7},{5}}
=> [3,7,4,6,5,1,2] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 5
{{1,3,4,6},{2},{5,7}}
=> [3,2,4,6,7,1,5] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,6},{2},{5},{7}}
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,7},{2,6},{5}}
=> [3,6,4,7,5,2,1] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => ? = 6
{{1,3,4,7},{2},{5,6}}
=> [3,2,4,7,6,5,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,7},{2},{5},{6}}
=> [3,2,4,7,5,6,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,5,6,7},{2,4}}
=> [3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 7
{{1,3,5,6},{2,4,7}}
=> [3,4,5,7,6,1,2] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,6},{2,4},{7}}
=> [3,4,5,2,6,1,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,7},{2,4,6}}
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,5,7},{2,4},{6}}
=> [3,4,5,2,7,6,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,6,7},{2,4,5}}
=> [3,4,6,5,2,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,6,7},{2,4},{5}}
=> [3,4,6,2,5,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,5,6,7},{2},{4}}
=> [3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 7
{{1,3,5,6},{2,7},{4}}
=> [3,7,5,4,6,1,2] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 6
{{1,3,5,6},{2},{4,7}}
=> [3,2,5,7,6,1,4] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,6},{2},{4},{7}}
=> [3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,7},{2,6},{4}}
=> [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => ? = 7
{{1,3,5,7},{2},{4,6}}
=> [3,2,5,6,7,4,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,5,7},{2},{4},{6}}
=> [3,2,5,4,7,6,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,6,7},{2,5},{4}}
=> [3,5,6,4,2,7,1] => [1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 8
{{1,3,6,7},{2},{4,5}}
=> [3,2,6,5,4,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,6,7},{2},{4},{5}}
=> [3,2,6,4,5,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,4,5,6,7},{2,3}}
=> [4,3,2,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 8
{{1,4,5,6},{2,3,7}}
=> [4,3,7,5,6,1,2] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 6
{{1,4,5,6},{2,3},{7}}
=> [4,3,2,5,6,1,7] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 6
{{1,4,5,7},{2,3,6}}
=> [4,3,6,5,7,2,1] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 7
{{1,4,5,7},{2,3},{6}}
=> [4,3,2,5,7,6,1] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 7
{{1,4,6,7},{2,3,5}}
=> [4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 8
{{1,4,6,7},{2,3},{5}}
=> [4,3,2,6,5,7,1] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 8
{{1,5,6,7},{2,3,4}}
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 9
{{1,5,6,7},{2,3},{4}}
=> [5,3,2,4,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 9
{{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 8
{{1,4,5,6},{2,7},{3}}
=> [4,7,3,5,6,1,2] => [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => ? = 7
Description
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
Matching statistic: St000803
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
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] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 5
{{1,3,4,5,6},{2,7}}
=> [3,7,4,5,6,1,2] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 4
{{1,3,4,5,6},{2},{7}}
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 4
{{1,3,4,5,7},{2,6}}
=> [3,6,4,5,7,2,1] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 5
{{1,3,4,5},{2,6,7}}
=> [3,6,4,5,1,7,2] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5},{2,6},{7}}
=> [3,6,4,5,1,2,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5,7},{2},{6}}
=> [3,2,4,5,7,6,1] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 5
{{1,3,4,5},{2,7},{6}}
=> [3,7,4,5,1,6,2] => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 4
{{1,3,4,5},{2},{6,7}}
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,5},{2},{6},{7}}
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3
{{1,3,4,6,7},{2,5}}
=> [3,5,4,6,2,7,1] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 6
{{1,3,4,6},{2,5,7}}
=> [3,5,4,6,7,1,2] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,6},{2,5},{7}}
=> [3,5,4,6,2,1,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,7},{2,5,6}}
=> [3,5,4,7,6,2,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,7},{2,5},{6}}
=> [3,5,4,7,2,6,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,6,7},{2},{5}}
=> [3,2,4,6,5,7,1] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 6
{{1,3,4,6},{2,7},{5}}
=> [3,7,4,6,5,1,2] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 5
{{1,3,4,6},{2},{5,7}}
=> [3,2,4,6,7,1,5] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,6},{2},{5},{7}}
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 4
{{1,3,4,7},{2,6},{5}}
=> [3,6,4,7,5,2,1] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => ? = 6
{{1,3,4,7},{2},{5,6}}
=> [3,2,4,7,6,5,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,4,7},{2},{5},{6}}
=> [3,2,4,7,5,6,1] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 5
{{1,3,5,6,7},{2,4}}
=> [3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 7
{{1,3,5,6},{2,4,7}}
=> [3,4,5,7,6,1,2] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,6},{2,4},{7}}
=> [3,4,5,2,6,1,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,7},{2,4,6}}
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,5,7},{2,4},{6}}
=> [3,4,5,2,7,6,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,6,7},{2,4,5}}
=> [3,4,6,5,2,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,6,7},{2,4},{5}}
=> [3,4,6,2,5,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,5,6,7},{2},{4}}
=> [3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 7
{{1,3,5,6},{2,7},{4}}
=> [3,7,5,4,6,1,2] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 6
{{1,3,5,6},{2},{4,7}}
=> [3,2,5,7,6,1,4] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,6},{2},{4},{7}}
=> [3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 5
{{1,3,5,7},{2,6},{4}}
=> [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => ? = 7
{{1,3,5,7},{2},{4,6}}
=> [3,2,5,6,7,4,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,5,7},{2},{4},{6}}
=> [3,2,5,4,7,6,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
{{1,3,6,7},{2,5},{4}}
=> [3,5,6,4,2,7,1] => [1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 8
{{1,3,6,7},{2},{4,5}}
=> [3,2,6,5,4,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,3,6,7},{2},{4},{5}}
=> [3,2,6,4,5,7,1] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 7
{{1,4,5,6,7},{2,3}}
=> [4,3,2,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 8
{{1,4,5,6},{2,3,7}}
=> [4,3,7,5,6,1,2] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 6
{{1,4,5,6},{2,3},{7}}
=> [4,3,2,5,6,1,7] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 6
{{1,4,5,7},{2,3,6}}
=> [4,3,6,5,7,2,1] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 7
{{1,4,5,7},{2,3},{6}}
=> [4,3,2,5,7,6,1] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 7
{{1,4,6,7},{2,3,5}}
=> [4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 8
{{1,4,6,7},{2,3},{5}}
=> [4,3,2,6,5,7,1] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 8
{{1,5,6,7},{2,3,4}}
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 9
{{1,5,6,7},{2,3},{4}}
=> [5,3,2,4,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 9
{{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 8
Description
The number of occurrences of the vincular pattern |132 in a permutation.
This is the number of occurrences of the pattern $(1,3,2)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St000795
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 57%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 57%
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] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,4,3,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => ? = 5
{{1,3,4,5,6},{2,7}}
=> [3,7,4,5,6,1,2] => [1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => ? = 4
{{1,3,4,5,6},{2},{7}}
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => ? = 4
{{1,3,4,5,7},{2,6}}
=> [3,6,4,5,7,2,1] => [1,3,4,5,7,2,6] => [1,7,5,4,3,2,6] => ? = 5
{{1,3,4,5},{2,6,7}}
=> [3,6,4,5,1,7,2] => [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
{{1,3,4,5},{2,6},{7}}
=> [3,6,4,5,1,2,7] => [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
{{1,3,4,5,7},{2},{6}}
=> [3,2,4,5,7,6,1] => [1,3,4,5,7,2,6] => [1,7,5,4,3,2,6] => ? = 5
{{1,3,4,5},{2,7},{6}}
=> [3,7,4,5,1,6,2] => [1,3,4,5,2,7,6] => [1,5,4,3,2,7,6] => ? = 4
{{1,3,4,5},{2},{6,7}}
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
{{1,3,4,5},{2},{6},{7}}
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
{{1,3,4,6,7},{2,5}}
=> [3,5,4,6,2,7,1] => [1,3,4,6,7,2,5] => [1,7,4,3,2,6,5] => ? = 6
{{1,3,4,6},{2,5,7}}
=> [3,5,4,6,7,1,2] => [1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => ? = 4
{{1,3,4,6},{2,5},{7}}
=> [3,5,4,6,2,1,7] => [1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => ? = 4
{{1,3,4,7},{2,5,6}}
=> [3,5,4,7,6,2,1] => [1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
{{1,3,4,7},{2,5},{6}}
=> [3,5,4,7,2,6,1] => [1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
{{1,3,4,6,7},{2},{5}}
=> [3,2,4,6,5,7,1] => [1,3,4,6,7,2,5] => [1,7,4,3,2,6,5] => ? = 6
{{1,3,4,6},{2,7},{5}}
=> [3,7,4,6,5,1,2] => [1,3,4,6,2,7,5] => [1,7,6,4,3,2,5] => ? = 5
{{1,3,4,6},{2},{5,7}}
=> [3,2,4,6,7,1,5] => [1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => ? = 4
{{1,3,4,6},{2},{5},{7}}
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => ? = 4
{{1,3,4,7},{2,6},{5}}
=> [3,6,4,7,5,2,1] => [1,3,4,7,2,6,5] => [1,6,7,4,3,2,5] => ? = 6
{{1,3,4,7},{2},{5,6}}
=> [3,2,4,7,6,5,1] => [1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
{{1,3,4,7},{2},{5},{6}}
=> [3,2,4,7,5,6,1] => [1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
{{1,3,5,6,7},{2,4}}
=> [3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,7,3,2,6,5,4] => ? = 7
{{1,3,5,6},{2,4,7}}
=> [3,4,5,7,6,1,2] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 5
{{1,3,5,6},{2,4},{7}}
=> [3,4,5,2,6,1,7] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 5
{{1,3,5,7},{2,4,6}}
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
{{1,3,5},{2,4,6,7}}
=> [3,4,5,6,1,7,2] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 3
{{1,3,5},{2,4,6},{7}}
=> [3,4,5,6,1,2,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 3
{{1,3,5,7},{2,4},{6}}
=> [3,4,5,2,7,6,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
{{1,3,5},{2,4,7},{6}}
=> [3,4,5,7,1,6,2] => [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 4
{{1,3,5},{2,4},{6,7}}
=> [3,4,5,2,1,7,6] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 3
{{1,3,5},{2,4},{6},{7}}
=> [3,4,5,2,1,6,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 3
{{1,3,6,7},{2,4,5}}
=> [3,4,6,5,2,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,6,4,5] => ? = 7
{{1,3,6},{2,4,5,7}}
=> [3,4,6,5,7,1,2] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 4
{{1,3,6},{2,4,5},{7}}
=> [3,4,6,5,2,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 4
{{1,3,7},{2,4,5,6}}
=> [3,4,7,5,6,2,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 5
{{1,3,7},{2,4,5},{6}}
=> [3,4,7,5,2,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 5
{{1,3,6,7},{2,4},{5}}
=> [3,4,6,2,5,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,6,4,5] => ? = 7
{{1,3,6},{2,4,7},{5}}
=> [3,4,6,7,5,1,2] => [1,3,6,2,4,7,5] => [1,7,6,3,2,4,5] => ? = 5
{{1,3,6},{2,4},{5,7}}
=> [3,4,6,2,7,1,5] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 4
{{1,3,6},{2,4},{5},{7}}
=> [3,4,6,2,5,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 4
{{1,3,7},{2,4,6},{5}}
=> [3,4,7,6,5,2,1] => [1,3,7,2,4,6,5] => [1,6,7,3,2,4,5] => ? = 6
{{1,3,7},{2,4},{5,6}}
=> [3,4,7,2,6,5,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 5
{{1,3,7},{2,4},{5},{6}}
=> [3,4,7,2,5,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 5
{{1,3,5,6,7},{2},{4}}
=> [3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,7,3,2,6,5,4] => ? = 7
{{1,3,5,6},{2,7},{4}}
=> [3,7,5,4,6,1,2] => [1,3,5,6,2,7,4] => [1,7,6,3,2,5,4] => ? = 6
{{1,3,5,6},{2},{4,7}}
=> [3,2,5,7,6,1,4] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 5
{{1,3,5,6},{2},{4},{7}}
=> [3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 5
{{1,3,5,7},{2,6},{4}}
=> [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,6,7,3,2,5,4] => ? = 7
Description
The mad of a permutation.
According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000081
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ? = 2
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [1,2,4,5,3,7,6] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ? = 2
{{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,4,6},{3,7},{5}}
=> [2,4,7,6,5,1,3] => [1,2,4,6,3,7,5] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 4
{{1,2,4,6},{3},{5,7}}
=> [2,4,3,6,7,1,5] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,4,6},{3},{5},{7}}
=> [2,4,3,6,5,1,7] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 5
{{1,2,4},{3,6,7},{5}}
=> [2,4,6,1,5,7,3] => [1,2,4,3,6,7,5] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
{{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ? = 2
{{1,2,4},{3,6},{5},{7}}
=> [2,4,6,1,5,3,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ? = 2
{{1,2,4,7},{3},{5,6}}
=> [2,4,3,7,6,5,1] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4},{3,7},{5,6}}
=> [2,4,7,1,6,5,3] => [1,2,4,3,7,5,6] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
{{1,2,4,7},{3},{5},{6}}
=> [2,4,3,7,5,6,1] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,4},{3,7},{5},{6}}
=> [2,4,7,1,5,6,3] => [1,2,4,3,7,5,6] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
{{1,2,4},{3},{5,7},{6}}
=> [2,4,3,1,7,6,5] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ? = 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [1,2,5,6,7,3,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [1,2,5,3,4,7,6] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
{{1,2,6,7},{3,4,5}}
=> [2,6,4,5,3,7,1] => [1,2,6,7,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
{{1,2,6,7},{3,4},{5}}
=> [2,6,4,3,5,7,1] => [1,2,6,7,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
{{1,2,6},{3,4,7},{5}}
=> [2,6,4,7,5,1,3] => [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [1,2,5,6,7,3,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [1,2,5,6,3,7,4] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [1,2,5,7,3,6,4] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 6
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [1,2,5,3,6,7,4] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 3
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [1,2,5,3,7,4,6] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [1,2,5,3,7,4,6] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 4
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [1,2,5,3,4,7,6] => ([(2,3),(4,6),(5,6)],7)
=> ? = 3
Description
The number of edges of a graph.
Matching statistic: St000246
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 64%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 64%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,1,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [5,4,2,3,1] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [5,3,2,1,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,4,2] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,4,2] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [5,3,4,1,2] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [5,2,1,4,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [5,2,4,3,1] => 2
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 3
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 3
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 3
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 4
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 4
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 3
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 4
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 3
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 3
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 4
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 2
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 2
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 4
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 3
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 2
{{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 2
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 5
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 3
{{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 3
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 4
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 4
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 2
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000961
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000961: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 71%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000961: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 71%
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] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,1,2] => 1
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [2,4,1,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [3,4,1,2] => 2
{{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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [3,4,1,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
{{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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [3,5,1,2,4] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [2,3,5,1,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [2,4,1,3,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [2,4,5,1,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [3,1,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [2,4,5,1,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [2,5,1,3,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [4,2,5,1,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [3,4,1,2,5] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [3,4,1,2,5] => 2
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 3
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 2
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 3
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 3
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 4
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 2
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 3
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 3
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => ? = 5
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 4
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 4
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ? = 2
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => ? = 5
{{1,2,4,6},{3,7},{5}}
=> [2,4,7,6,5,1,3] => [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => ? = 4
{{1,2,4,7},{3},{5,6}}
=> [2,4,3,7,6,5,1] => [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 4
{{1,2,4},{3},{5,6,7}}
=> [2,4,3,1,6,7,5] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4},{3},{5,6},{7}}
=> [2,4,3,1,6,5,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
{{1,2,4,7},{3},{5},{6}}
=> [2,4,3,7,5,6,1] => [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 4
Description
The shifted major index of a permutation.
This is given by the sum of all indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
Summing with [[St000354]] yields Rawlings' Mahonian statistic, see [1, p. 50].
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001397Number of pairs of incomparable elements in a finite poset. St001511The minimal number of transpositions needed to sort a permutation in either direction. St000004The major index of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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