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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000154
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000154: 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] => [2,1] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,2,1] => 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 5
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 4
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [2,4,3,1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [3,2,4,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [3,4,2,1] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,4,5,3] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => 7
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,3,4,2,5] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,4,2] => 6
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => 5
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,4,5,3,2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,4,3,2] => 9
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [3,5,1,2,4] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 5
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [4,5,1,2,3] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [5,4,1,2,3] => 5
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => 4
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [4,1,2,5,3] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,4,5,3] => 4
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,5,4,3] => 8
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [2,5,1,3,4] => 1
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
Matching statistic: St000492
(load all 461 compositions to match this statistic)
(load all 461 compositions to match this statistic)
St000492: 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}}
=> 1
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 5
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 4
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 3
{{1},{2,4},{3}}
=> 3
{{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 3
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 7
{{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 6
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 6
{{1,2,5},{3},{4}}
=> 5
{{1,2},{3,5},{4}}
=> 5
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 9
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 5
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 5
{{1,3,5},{2},{4}}
=> 4
{{1,3},{2,5},{4}}
=> 4
{{1,3},{2},{4,5}}
=> 4
{{1,3},{2},{4},{5}}
=> 8
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 5
Description
The rob 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 '''rob''' (right-opener-bigger) 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 is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.
Matching statistic: St000577
(load all 209 compositions to match this statistic)
(load all 209 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
St000577: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000577: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1},{2,3}}
=> 2
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 3
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 5
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 4
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 3
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 3
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 4
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 3
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 3
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 7
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 2
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 6
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 2
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 6
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 5
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 5
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 9
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 5
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 5
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 4
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 4
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 4
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 8
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 5
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block.
Matching statistic: St000472
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000472: 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] => 1
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
=> {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{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] => 3
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 3
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 2
{{1,2},{3,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 2
{{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 5
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 4
{{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 1
{{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 4
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 3
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 3
{{1},{2},{3,4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 3
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 6
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => 4
{{1,2,3,5},{4}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => 3
{{1,2,3},{4,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => 3
{{1,2,3},{4},{5}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => 7
{{1,2,4,5},{3}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => 2
{{1,2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => 2
{{1,2,4},{3},{5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => 6
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => 2
{{1,2},{3,4,5}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,3,2,5,1] => 2
{{1,2},{3,4},{5}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => 6
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,4,1] => 5
{{1,2},{3,5},{4}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => 5
{{1,2},{3},{4,5}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => 5
{{1,2},{3},{4},{5}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => 9
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,4,3,2] => 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,3,2] => 5
{{1,3,5},{2,4}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => 1
{{1,3},{2,4,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => 1
{{1,3},{2,4},{5}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => 5
{{1,3,5},{2},{4}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,4,2] => 4
{{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,5,1,4,2] => 4
{{1,3},{2},{4,5}}
=> {{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => 4
{{1,3},{2},{4},{5}}
=> {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => 8
{{1,4,5},{2,3}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
{{1,4},{2,3,5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 5
Description
The sum of the ascent bottoms of a permutation.
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