Your data matches 29 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001727
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => 0
{{1,2}}
 => [2,1] => 0
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
{{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 $\pi(i) > \pi(j) > i$.
Matching statistic: St000356
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000356: 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] => 1
{{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] => 1
{{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] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => 2
{{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] => 2
{{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] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000223
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: 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] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [3,2,1] => 1
{{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] => [4,1,2,3] => [4,1,2,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [4,1,3,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [4,2,1,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [4,3,2,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [4,2,3,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1
{{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] => [5,1,2,3,4] => [5,1,2,3,4] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [5,1,2,4,3] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [5,1,3,2,4] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [5,1,4,2,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [4,1,3,2,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [5,1,4,3,2] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [5,1,3,4,2] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [5,2,1,3,4] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [5,4,1,3,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [5,4,2,1,3] => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [5,3,1,2,4] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [5,2,1,4,3] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [5,3,1,4,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [5,3,2,1,4] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [5,4,1,2,3] => 1
Description
The number of nestings in the permutation.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: 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] => [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] => 1
{{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] => 1
{{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] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,4,2] => 2
{{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] => 2
{{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] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [5,2,4,3,1] => 2
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000491
Mapping
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 2
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 1
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 2
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 3
{{1,3,4},{2,5}}
 => 2
{{1,3,4},{2},{5}}
 => 2
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => 3
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 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.
Matching statistic: St000497
Mapping
St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 2
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 2
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 2
{{1,4},{2,3},{5}}
 => 2
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: St000565
Mapping
St000565: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 1
{{1},{2,4},{3}}
 => 2
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 2
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 1
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 2
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 3
{{1,3,4},{2,5}}
 => 2
{{1,3,4},{2},{5}}
 => 2
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => 3
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
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: St000609
(load all 2 compositions to match this statistic)
Mapping
St000609: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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.
Matching statistic: St000589
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000809The reduced reflection length of the permutation. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001964The interval resolution global dimension of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000516The number of stretching pairs of a permutation. St001438The number of missing boxes of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001435The number of missing boxes in the first row. St001867The number of alignments of type EN of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car.