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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000355
Mp00112: Set partitions —complement⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000355: 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}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 0
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 0
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 0
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 3
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 0
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 2
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 0
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 2
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => 0
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 0
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 0
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 0
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 3
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 1
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000597
(load all 82 compositions to match this statistic)
(load all 82 compositions to match this statistic)
St000597: 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}}
=> 2
{{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}}
=> 1
{{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}}
=> 3
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 3
{{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}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000556
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000556: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000556: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{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] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{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] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} in a set partition.
Matching statistic: St000586
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{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] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{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] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Matching statistic: St000599
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{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] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{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] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block.
Matching statistic: St000605
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{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] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{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] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000607
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000607: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000607: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{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] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{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] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block.
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