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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000356
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%
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: St001727
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
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}}
=> {{1}}
=> [1] => 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [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}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 2
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 2
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation π is a pair i<j such that π(j)≤min. Thus, an invisible inversion satisfies \pi(i) > \pi(j) > i.
Matching statistic: St000497
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: St000491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> 1
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,3,4},{2,5}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,4},{2},{3,5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1,3},{2,5},{4}}
=> 2
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> {{1,2,4,5},{3}}
=> 2
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4,5},{2,3}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,4,5},{3}}
=> 2
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: St000609
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 1
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> 1
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> 0
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 0
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> 1
{{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> 2
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> 2
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000809
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: 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] => [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] => 1
{{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] => 1
{{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] => [1,5,3,4,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{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] => 2
{{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] => 2
{{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
Description
The reduced reflection length of the permutation.
Let T be the set of reflections in a Coxeter group and let \ell(w) be the usual length function. Then the reduced reflection length of w is
\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.
In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St000358
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●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
{{1},{2,3,4,5,6,7}}
=> [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4,5,7},{6}}
=> [1,3,4,5,7,6,2] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4,6,7},{5}}
=> [1,3,4,6,5,7,2] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
{{1},{2,3,4,6},{5,7}}
=> [1,3,4,6,7,2,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1},{2,3,4},{5,6,7}}
=> [1,3,4,2,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4},{5,6},{7}}
=> [1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1},{2,3,4},{5,7},{6}}
=> [1,3,4,2,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,5,6,7},{4}}
=> [1,3,5,4,6,7,2] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 1
{{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
{{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2
{{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2
{{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 2
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
{{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2
{{1},{2,3,6},{4,5,7}}
=> [1,3,6,5,7,2,4] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1},{2,3},{4,5,6,7}}
=> [1,3,2,5,6,7,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3},{4,5,6},{7}}
=> [1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,7},{4,5},{6}}
=> [1,3,7,5,4,6,2] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1},{2,3},{4,5,7},{6}}
=> [1,3,2,5,7,6,4] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
{{1},{2,3},{4,5},{6,7}}
=> [1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3},{4,5},{6},{7}}
=> [1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,6,7},{4},{5}}
=> [1,3,6,4,5,7,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2
{{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 3
{{1},{2,3,6},{4},{5,7}}
=> [1,3,6,4,7,2,5] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
{{1},{2,3,7},{4,6},{5}}
=> [1,3,7,6,5,4,2] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 4
{{1},{2,3},{4,6,7},{5}}
=> [1,3,2,6,5,7,4] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
{{1},{2,3},{4,6},{5,7}}
=> [1,3,2,6,7,4,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1},{2,3},{4,6},{5},{7}}
=> [1,3,2,6,5,4,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
{{1},{2,3,7},{4},{5,6}}
=> [1,3,7,4,6,5,2] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1},{2,3},{4,7},{5,6}}
=> [1,3,2,7,6,5,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1},{2,3},{4},{5,6,7}}
=> [1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
{{1},{2,3,7},{4},{5},{6}}
=> [1,3,7,4,5,6,2] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
{{1},{2,3},{4,7},{5},{6}}
=> [1,3,2,7,5,6,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
{{1},{2,3},{4},{5,7},{6}}
=> [1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
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: St001905
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 43%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 43%
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] => [1,3,4,2] => 1
{{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] => [1,2,4,5,3] => ? = 1
{{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] => [1,3,4,5,2] => ? = 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
{{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] => [1,3,5,2,4] => ? = 2
{{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] => ? = 2
{{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
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 4
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let (a_1,\dots,a_n) be a parking function. Then this statistic returns the number of indices 1\leq i\leq n such that a_i < i.
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