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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000807
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0
{{1,2}}
=> [2] => 0
{{1},{2}}
=> [1,1] => 0
{{1,2,3}}
=> [3] => 0
{{1,2},{3}}
=> [2,1] => 0
{{1,3},{2}}
=> [2,1] => 0
{{1},{2,3}}
=> [1,2] => 0
{{1},{2},{3}}
=> [1,1,1] => 0
{{1,2,3,4}}
=> [4] => 0
{{1,2,3},{4}}
=> [3,1] => 0
{{1,2,4},{3}}
=> [3,1] => 0
{{1,2},{3,4}}
=> [2,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => 0
{{1,3,4},{2}}
=> [3,1] => 0
{{1,3},{2,4}}
=> [2,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => 0
{{1,4},{2,3}}
=> [2,2] => 0
{{1},{2,3,4}}
=> [1,3] => 0
{{1},{2,3},{4}}
=> [1,2,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => 0
{{1},{2,4},{3}}
=> [1,2,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => 0
{{1,2,3,4,5}}
=> [5] => 0
{{1,2,3,4},{5}}
=> [4,1] => 0
{{1,2,3,5},{4}}
=> [4,1] => 0
{{1,2,3},{4,5}}
=> [3,2] => 0
{{1,2,3},{4},{5}}
=> [3,1,1] => 0
{{1,2,4,5},{3}}
=> [4,1] => 0
{{1,2,4},{3,5}}
=> [3,2] => 0
{{1,2,4},{3},{5}}
=> [3,1,1] => 0
{{1,2,5},{3,4}}
=> [3,2] => 0
{{1,2},{3,4,5}}
=> [2,3] => 0
{{1,2},{3,4},{5}}
=> [2,2,1] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => 0
{{1,2},{3,5},{4}}
=> [2,2,1] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 0
{{1,3,4,5},{2}}
=> [4,1] => 0
{{1,3,4},{2,5}}
=> [3,2] => 0
{{1,3,4},{2},{5}}
=> [3,1,1] => 0
{{1,3,5},{2,4}}
=> [3,2] => 0
{{1,3},{2,4,5}}
=> [2,3] => 0
{{1,3},{2,4},{5}}
=> [2,2,1] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => 0
{{1,3},{2,5},{4}}
=> [2,2,1] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 0
{{1,4,5},{2,3}}
=> [3,2] => 0
{{1,4},{2,3,5}}
=> [2,3] => 0
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St001771
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{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] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 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] => [4,1,3,2] => [4,1,3,2] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 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] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1
{{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] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,3,6,5,2,4] => [1,3,6,5,2,4] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4,6] => ? = 0
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{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] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 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] => [4,1,3,2] => [4,1,3,2] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 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] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1
{{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] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,3,6,5,2,4] => [1,3,6,5,2,4] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4,6] => ? = 0
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{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] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 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] => [4,1,3,2] => [4,1,3,2] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 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] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1
{{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] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,3,6,5,2,4] => [1,3,6,5,2,4] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4,6] => ? = 0
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000068
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ([(0,3),(0,4),(0,5),(1,8),(1,14),(2,6),(2,7),(3,10),(3,11),(4,2),(4,11),(4,12),(5,1),(5,10),(5,12),(6,13),(6,15),(7,13),(7,15),(8,13),(8,15),(10,14),(11,7),(11,14),(12,6),(12,8),(12,14),(13,9),(14,15),(15,9)],16)
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1,3},{2},{4},{5},{6}}
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
{{1,4},{2,3},{5},{6}}
=> [4,3,2,1,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
{{1,5},{2,4},{3},{6}}
=> [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
{{1},{2,6},{3,5},{4}}
=> [1,6,5,4,3,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1},{2},{3,6},{4,5}}
=> [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
{{1},{2},{3},{4,6},{5}}
=> [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
{{1,7},{2,6},{3,5},{4}}
=> [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
{{1,8},{2,7},{3,6},{4,5}}
=> [8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001862
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{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] => [4,2,3,1] => [4,2,3,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0
{{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] => [4,2,3,1] => [4,2,3,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 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] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{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] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 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] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1
{{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] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => [5,2,3,4,1,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => [4,2,3,1,6,5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => [4,2,3,1,5,6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [6,2,4,3,5,1] => [6,2,4,3,5,1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => [5,2,6,4,1,3] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,4,3,1,6] => [5,2,4,3,1,6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 0
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001772
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [2,3,1] => [-2,-3,-1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
{{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] => [2,3,4,1] => [-2,-3,-4,-1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
{{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] => [3,2,4,1] => [-3,-2,-4,-1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 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] => [4,2,3,1] => [-4,-2,-3,-1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
{{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] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 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] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1
{{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] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-2,-3,-4,-5,-6,-1] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => [-2,-3,-4,-5,-1,-6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => [-2,-3,-4,-6,-5,-1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => [-2,-3,-4,-1,-6,-5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => [-2,-3,-4,-1,-5,-6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-2,-3,-5,-4,-6,-1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => [-2,-3,-5,-6,-1,-4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => [-2,-3,-5,-4,-1,-6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-2,-3,-6,-5,-4,-1] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,3,1,5,6,4] => [-2,-3,-1,-5,-6,-4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => [-2,-3,-1,-5,-4,-6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-2,-3,-6,-4,-5,-1] => ? = 0
Description
The number of occurrences of the signed pattern 12 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < \pi(j)$.
Matching statistic: St001863
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [2,3,1] => [-2,-3,-1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
{{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] => [2,3,4,1] => [-2,-3,-4,-1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
{{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] => [3,2,4,1] => [-3,-2,-4,-1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 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] => [4,2,3,1] => [-4,-2,-3,-1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
{{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] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 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] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1
{{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] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-2,-3,-4,-5,-6,-1] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => [-2,-3,-4,-5,-1,-6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => [-2,-3,-4,-6,-5,-1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => [-2,-3,-4,-1,-6,-5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => [-2,-3,-4,-1,-5,-6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-2,-3,-5,-4,-6,-1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => [-2,-3,-5,-6,-1,-4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => [-2,-3,-5,-4,-1,-6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-2,-3,-6,-5,-4,-1] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,3,1,5,6,4] => [-2,-3,-1,-5,-6,-4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => [-2,-3,-1,-5,-4,-6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-2,-3,-6,-4,-5,-1] => ? = 0
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Matching statistic: St001864
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [2,3,1] => [-2,-3,-1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
{{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] => [2,3,4,1] => [-2,-3,-4,-1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
{{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] => [3,2,4,1] => [-3,-2,-4,-1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 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] => [4,2,3,1] => [-4,-2,-3,-1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
{{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] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 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] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1
{{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] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-2,-3,-4,-5,-6,-1] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => [-2,-3,-4,-5,-1,-6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => [-2,-3,-4,-6,-5,-1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => [-2,-3,-4,-1,-6,-5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => [-2,-3,-4,-1,-5,-6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-2,-3,-5,-4,-6,-1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => [-2,-3,-5,-6,-1,-4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => [-2,-3,-5,-4,-1,-6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-2,-3,-6,-5,-4,-1] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,3,1,5,6,4] => [-2,-3,-1,-5,-6,-4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => [-2,-3,-1,-5,-4,-6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-2,-3,-6,-4,-5,-1] => ? = 0
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001867
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
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] => [2,3,1] => [-2,-3,-1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
{{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] => [2,3,4,1] => [-2,-3,-4,-1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
{{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] => [3,2,4,1] => [-3,-2,-4,-1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 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] => [4,2,3,1] => [-4,-2,-3,-1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
{{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] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 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] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1
{{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] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-2,-3,-4,-5,-6,-1] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => [-2,-3,-4,-5,-1,-6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => [-2,-3,-4,-6,-5,-1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => [-2,-3,-4,-1,-6,-5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => [-2,-3,-4,-1,-5,-6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-2,-3,-5,-4,-6,-1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => [-2,-3,-5,-6,-1,-4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => [-2,-3,-5,-4,-1,-6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-2,-3,-6,-5,-4,-1] => ? = 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,3,1,5,6,4] => [-2,-3,-1,-5,-6,-4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => [-2,-3,-1,-5,-4,-6] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-2,-3,-6,-4,-5,-1] => ? = 0
Description
The number of alignments of type EN of a signed permutation.
An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:
* $-i < 0 < -\pi(i) < \pi(j) < j$
* $i \leq\pi(i) < \pi(j) < j$.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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