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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000708
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St001864
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,1,2] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,5,3] => [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,5,2] => [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,5,3] => [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,3,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,3,4,2] => [1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,4,5,3] => [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,3,4,5,1] => [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,3,5,4,1] => [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,4,3,5,1] => [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,4,5,3,1] => [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,5,3,4,1] => [1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[3,2,4,5,1] => [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,4,1,2,5] => [2,4,1,3,5] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[3,4,1,5,2] => [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,4,5,1,2] => [1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[4,2,3,1,5] => [4,1,3,2,5] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
[4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[4,5,1,3,2] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[4,5,2,1,3] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 2
[4,5,3,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[5,1,2,3,4] => [5,1,2,3,4] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,2,4,3] => [5,1,2,4,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,3,2,4] => [5,1,3,2,4] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,3,4,2] => [5,1,2,4,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,4,2,3] => [5,1,4,2,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,2,1,3,4] => [5,2,1,3,4] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[5,2,3,4,1] => [5,1,2,4,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 2
[5,3,4,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,4,2,3,1] => [5,4,1,3,2] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,4,6,5,3] => [1,2,3,6,5,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
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: St001769
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,3,4,5,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[2,3,5,4,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[2,4,3,5,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[2,4,5,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[2,5,3,4,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[2,5,4,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[3,2,4,5,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1
[3,4,1,2,5] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,4,1,5,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,4,5,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
[3,5,1,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,5,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
[4,2,3,1,5] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[4,5,1,2,3] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[4,5,1,3,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[4,5,2,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 2
[4,5,3,1,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[5,1,2,3,4] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,2,4,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,3,2,4] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,3,4,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,4,2,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[5,2,1,3,4] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[5,2,3,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[5,3,2,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[5,3,4,2,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[5,4,2,3,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 2
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
Matching statistic: St001896
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 2
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,2,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[2,1,3,4,5] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,1,3,5,4] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,1,4,3,5] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[2,3,4,5,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,3,5,4,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,4,3,5,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,4,5,3,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,5,3,4,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,5,4,3,1] => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[3,2,1,4,5] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[3,2,4,5,1] => [3,2,5,4,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[3,4,1,2,5] => [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[3,4,1,5,2] => [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[3,4,5,1,2] => [3,5,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
[3,5,1,2,4] => [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[3,5,1,4,2] => [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[3,5,4,1,2] => [3,5,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
[4,2,3,1,5] => [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
[4,3,2,5,1] => [4,3,2,5,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[4,3,5,1,2] => [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 2
[4,5,1,2,3] => [4,5,1,3,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[4,5,1,3,2] => [4,5,1,3,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[4,5,2,1,3] => [4,5,2,1,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2
[4,5,3,1,2] => [4,5,3,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[5,1,2,3,4] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,1,2,4,3] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,1,3,2,4] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,1,3,4,2] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,1,4,2,3] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,1,4,3,2] => [5,1,4,3,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[5,2,1,3,4] => [5,2,1,4,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[5,2,1,4,3] => [5,2,1,4,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[5,2,3,4,1] => [5,2,4,3,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 2
[5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 2
[5,3,2,1,4] => [5,3,2,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[5,3,4,2,1] => [5,3,4,2,1] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[5,4,2,3,1] => [5,4,2,3,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,6,2] => ? = 2
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,6,2] => ? = 2
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,6,2] => ? = 2
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,6,2] => ? = 2
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St000327
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [2,3,4,1] => [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 + 2
[1,2,4,3] => [2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,3,2,4] => [3,2,4,1] => [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 + 2
[1,3,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,2,3] => [3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 2
[3,4,1,2] => [4,1,2,3] => [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)
=> ? = 2 + 2
[4,2,3,1] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 2
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,2,3,4,5] => [2,3,4,5,1] => [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)
=> ? = 2 + 2
[1,2,3,5,4] => [2,3,5,4,1] => [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)
=> ? = 2 + 2
[1,2,4,3,5] => [2,4,3,5,1] => [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)
=> ? = 2 + 2
[1,2,4,5,3] => [2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,4,3] => [2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,2,4,5] => [3,2,4,5,1] => [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)
=> ? = 2 + 2
[1,3,2,5,4] => [3,2,5,4,1] => [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)
=> ? = 2 + 2
[1,3,4,2,5] => [4,2,3,5,1] => [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)
=> ? = 2 + 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,2,3,5] => [3,4,2,5,1] => [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)
=> ? = 2 + 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,3,2,5] => [4,3,2,5,1] => [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)
=> ? = 2 + 2
[1,4,3,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,2,3] => [4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,3,4] => [3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,4,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,2,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,4,2] => [5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,2,3] => [4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [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)
=> ? = 2 + 2
[2,1,3,5,4] => [1,3,5,4,2] => [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)
=> ? = 2 + 2
[2,1,4,3,5] => [1,4,3,5,2] => [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)
=> ? = 2 + 2
[2,1,4,5,3] => [1,5,3,4,2] => [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)
=> ? = 2 + 2
[2,1,5,3,4] => [1,4,5,3,2] => [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)
=> ? = 2 + 2
[2,1,5,4,3] => [1,5,4,3,2] => [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)
=> ? = 2 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,3,5,4,1] => [1,2,4,3,5] => [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)
=> ? = 2 + 2
[2,4,3,5,1] => [1,3,2,4,5] => [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)
=> ? = 2 + 2
[2,4,5,3,1] => [1,4,2,3,5] => [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)
=> ? = 2 + 2
[2,5,3,4,1] => [1,3,4,2,5] => [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)
=> ? = 2 + 2
[2,5,4,3,1] => [1,4,3,2,5] => [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)
=> ? = 2 + 2
[3,2,1,4,5] => [2,1,4,5,3] => [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)
=> ? = 2 + 2
[3,2,1,5,4] => [2,1,5,4,3] => [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)
=> ? = 2 + 2
[3,2,4,5,1] => [2,1,3,4,5] => [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 + 2
[3,2,5,4,1] => [2,1,4,3,5] => [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)
=> ? = 1 + 2
[3,4,1,2,5] => [4,1,2,5,3] => [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)
=> ? = 2 + 2
[3,4,1,5,2] => [5,1,2,4,3] => [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)
=> ? = 2 + 2
[3,4,5,1,2] => [5,1,2,3,4] => [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)
=> ? = 2 + 2
[3,5,1,2,4] => [4,1,5,2,3] => [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)
=> ? = 2 + 2
[3,5,1,4,2] => [5,1,4,2,3] => [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)
=> ? = 2 + 2
[3,5,4,1,2] => [5,1,3,2,4] => [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)
=> ? = 2 + 2
[4,2,3,1,5] => [2,3,1,5,4] => [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)
=> ? = 2 + 2
[4,2,5,1,3] => [2,5,1,3,4] => [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)
=> ? = 2 + 2
[4,3,2,1,5] => [3,2,1,5,4] => [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)
=> ? = 2 + 2
[4,3,2,5,1] => [3,2,1,4,5] => [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)
=> ? = 2 + 2
[4,3,5,1,2] => [5,2,1,3,4] => [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)
=> ? = 2 + 2
[4,5,1,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,1,3,2] => [5,4,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,2,1,3] => [3,5,1,2,4] => [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)
=> ? = 2 + 2
[4,5,3,1,2] => [5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,3,4] => [3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,4,3] => [3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,2,4] => [4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,4,2] => [5,3,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,2,3] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,3,2] => [5,4,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,2,1,3,4] => [2,4,5,1,3] => [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)
=> ? = 1 + 2
[5,2,1,4,3] => [2,5,4,1,3] => [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)
=> ? = 1 + 2
[5,2,3,4,1] => [2,3,4,1,5] => [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)
=> ? = 2 + 2
[5,2,4,3,1] => [2,4,3,1,5] => [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)
=> ? = 2 + 2
[5,3,2,1,4] => [3,2,5,1,4] => [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)
=> ? = 2 + 2
[5,3,2,4,1] => [3,2,4,1,5] => [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)
=> ? = 2 + 2
[5,3,4,2,1] => [4,2,3,1,5] => [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)
=> ? = 2 + 2
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St001637
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [2,3,4,1] => [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 + 2
[1,2,4,3] => [2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,3,2,4] => [3,2,4,1] => [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 + 2
[1,3,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,2,3] => [3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 2
[3,4,1,2] => [4,1,2,3] => [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)
=> ? = 2 + 2
[4,2,3,1] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 2
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,2,3,4,5] => [2,3,4,5,1] => [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)
=> ? = 2 + 2
[1,2,3,5,4] => [2,3,5,4,1] => [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)
=> ? = 2 + 2
[1,2,4,3,5] => [2,4,3,5,1] => [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)
=> ? = 2 + 2
[1,2,4,5,3] => [2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,4,3] => [2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,2,4,5] => [3,2,4,5,1] => [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)
=> ? = 2 + 2
[1,3,2,5,4] => [3,2,5,4,1] => [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)
=> ? = 2 + 2
[1,3,4,2,5] => [4,2,3,5,1] => [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)
=> ? = 2 + 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,2,3,5] => [3,4,2,5,1] => [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)
=> ? = 2 + 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,3,2,5] => [4,3,2,5,1] => [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)
=> ? = 2 + 2
[1,4,3,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,2,3] => [4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,3,4] => [3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,4,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,2,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,4,2] => [5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,2,3] => [4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [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)
=> ? = 2 + 2
[2,1,3,5,4] => [1,3,5,4,2] => [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)
=> ? = 2 + 2
[2,1,4,3,5] => [1,4,3,5,2] => [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)
=> ? = 2 + 2
[2,1,4,5,3] => [1,5,3,4,2] => [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)
=> ? = 2 + 2
[2,1,5,3,4] => [1,4,5,3,2] => [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)
=> ? = 2 + 2
[2,1,5,4,3] => [1,5,4,3,2] => [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)
=> ? = 2 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,3,5,4,1] => [1,2,4,3,5] => [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)
=> ? = 2 + 2
[2,4,3,5,1] => [1,3,2,4,5] => [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)
=> ? = 2 + 2
[2,4,5,3,1] => [1,4,2,3,5] => [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)
=> ? = 2 + 2
[2,5,3,4,1] => [1,3,4,2,5] => [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)
=> ? = 2 + 2
[2,5,4,3,1] => [1,4,3,2,5] => [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)
=> ? = 2 + 2
[3,2,1,4,5] => [2,1,4,5,3] => [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)
=> ? = 2 + 2
[3,2,1,5,4] => [2,1,5,4,3] => [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)
=> ? = 2 + 2
[3,2,4,5,1] => [2,1,3,4,5] => [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 + 2
[3,2,5,4,1] => [2,1,4,3,5] => [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)
=> ? = 1 + 2
[3,4,1,2,5] => [4,1,2,5,3] => [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)
=> ? = 2 + 2
[3,4,1,5,2] => [5,1,2,4,3] => [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)
=> ? = 2 + 2
[3,4,5,1,2] => [5,1,2,3,4] => [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)
=> ? = 2 + 2
[3,5,1,2,4] => [4,1,5,2,3] => [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)
=> ? = 2 + 2
[3,5,1,4,2] => [5,1,4,2,3] => [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)
=> ? = 2 + 2
[3,5,4,1,2] => [5,1,3,2,4] => [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)
=> ? = 2 + 2
[4,2,3,1,5] => [2,3,1,5,4] => [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)
=> ? = 2 + 2
[4,2,5,1,3] => [2,5,1,3,4] => [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)
=> ? = 2 + 2
[4,3,2,1,5] => [3,2,1,5,4] => [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)
=> ? = 2 + 2
[4,3,2,5,1] => [3,2,1,4,5] => [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)
=> ? = 2 + 2
[4,3,5,1,2] => [5,2,1,3,4] => [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)
=> ? = 2 + 2
[4,5,1,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,1,3,2] => [5,4,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,2,1,3] => [3,5,1,2,4] => [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)
=> ? = 2 + 2
[4,5,3,1,2] => [5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,3,4] => [3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,4,3] => [3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,2,4] => [4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,4,2] => [5,3,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,2,3] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,3,2] => [5,4,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,2,1,3,4] => [2,4,5,1,3] => [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)
=> ? = 1 + 2
[5,2,1,4,3] => [2,5,4,1,3] => [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)
=> ? = 1 + 2
[5,2,3,4,1] => [2,3,4,1,5] => [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)
=> ? = 2 + 2
[5,2,4,3,1] => [2,4,3,1,5] => [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)
=> ? = 2 + 2
[5,3,2,1,4] => [3,2,5,1,4] => [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)
=> ? = 2 + 2
[5,3,2,4,1] => [3,2,4,1,5] => [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)
=> ? = 2 + 2
[5,3,4,2,1] => [4,2,3,1,5] => [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)
=> ? = 2 + 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [2,3,4,1] => [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 + 2
[1,2,4,3] => [2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,3,2,4] => [3,2,4,1] => [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 + 2
[1,3,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,2,3] => [3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 2
[3,4,1,2] => [4,1,2,3] => [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)
=> ? = 2 + 2
[4,2,3,1] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 2
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,2,3,4,5] => [2,3,4,5,1] => [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)
=> ? = 2 + 2
[1,2,3,5,4] => [2,3,5,4,1] => [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)
=> ? = 2 + 2
[1,2,4,3,5] => [2,4,3,5,1] => [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)
=> ? = 2 + 2
[1,2,4,5,3] => [2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,2,5,4,3] => [2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,2,4,5] => [3,2,4,5,1] => [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)
=> ? = 2 + 2
[1,3,2,5,4] => [3,2,5,4,1] => [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)
=> ? = 2 + 2
[1,3,4,2,5] => [4,2,3,5,1] => [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)
=> ? = 2 + 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,3,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,2,3,5] => [3,4,2,5,1] => [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)
=> ? = 2 + 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,3,2,5] => [4,3,2,5,1] => [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)
=> ? = 2 + 2
[1,4,3,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,2,3] => [4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,3,4] => [3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,2,4,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,2,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,3,4,2] => [5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,2,3] => [4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [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)
=> ? = 2 + 2
[2,1,3,5,4] => [1,3,5,4,2] => [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)
=> ? = 2 + 2
[2,1,4,3,5] => [1,4,3,5,2] => [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)
=> ? = 2 + 2
[2,1,4,5,3] => [1,5,3,4,2] => [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)
=> ? = 2 + 2
[2,1,5,3,4] => [1,4,5,3,2] => [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)
=> ? = 2 + 2
[2,1,5,4,3] => [1,5,4,3,2] => [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)
=> ? = 2 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[2,3,5,4,1] => [1,2,4,3,5] => [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)
=> ? = 2 + 2
[2,4,3,5,1] => [1,3,2,4,5] => [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)
=> ? = 2 + 2
[2,4,5,3,1] => [1,4,2,3,5] => [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)
=> ? = 2 + 2
[2,5,3,4,1] => [1,3,4,2,5] => [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)
=> ? = 2 + 2
[2,5,4,3,1] => [1,4,3,2,5] => [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)
=> ? = 2 + 2
[3,2,1,4,5] => [2,1,4,5,3] => [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)
=> ? = 2 + 2
[3,2,1,5,4] => [2,1,5,4,3] => [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)
=> ? = 2 + 2
[3,2,4,5,1] => [2,1,3,4,5] => [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 + 2
[3,2,5,4,1] => [2,1,4,3,5] => [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)
=> ? = 1 + 2
[3,4,1,2,5] => [4,1,2,5,3] => [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)
=> ? = 2 + 2
[3,4,1,5,2] => [5,1,2,4,3] => [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)
=> ? = 2 + 2
[3,4,5,1,2] => [5,1,2,3,4] => [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)
=> ? = 2 + 2
[3,5,1,2,4] => [4,1,5,2,3] => [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)
=> ? = 2 + 2
[3,5,1,4,2] => [5,1,4,2,3] => [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)
=> ? = 2 + 2
[3,5,4,1,2] => [5,1,3,2,4] => [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)
=> ? = 2 + 2
[4,2,3,1,5] => [2,3,1,5,4] => [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)
=> ? = 2 + 2
[4,2,5,1,3] => [2,5,1,3,4] => [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)
=> ? = 2 + 2
[4,3,2,1,5] => [3,2,1,5,4] => [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)
=> ? = 2 + 2
[4,3,2,5,1] => [3,2,1,4,5] => [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)
=> ? = 2 + 2
[4,3,5,1,2] => [5,2,1,3,4] => [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)
=> ? = 2 + 2
[4,5,1,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,1,3,2] => [5,4,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[4,5,2,1,3] => [3,5,1,2,4] => [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)
=> ? = 2 + 2
[4,5,3,1,2] => [5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,3,4] => [3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,2,4,3] => [3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,2,4] => [4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,3,4,2] => [5,3,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,2,3] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,1,4,3,2] => [5,4,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
[5,2,1,3,4] => [2,4,5,1,3] => [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)
=> ? = 1 + 2
[5,2,1,4,3] => [2,5,4,1,3] => [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)
=> ? = 1 + 2
[5,2,3,4,1] => [2,3,4,1,5] => [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)
=> ? = 2 + 2
[5,2,4,3,1] => [2,4,3,1,5] => [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)
=> ? = 2 + 2
[5,3,2,1,4] => [3,2,5,1,4] => [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)
=> ? = 2 + 2
[5,3,2,4,1] => [3,2,4,1,5] => [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)
=> ? = 2 + 2
[5,3,4,2,1] => [4,2,3,1,5] => [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)
=> ? = 2 + 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000454
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 18%
Values
[1,2,3,4] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,2,4,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,3,2,4] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,5,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,3,5,4,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,4,3,5,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,4,5,3,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,5,3,4,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[3,2,1,4,5] => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,5,1] => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[6,2,3,4,5,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,3,4,7,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,3,5,4,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,3,5,7,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,3,7,4,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,3,7,5,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,3,5,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,3,7,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,5,3,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,5,7,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,7,3,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,4,7,5,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,3,4,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,3,7,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,4,3,1,7] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,4,7,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,7,3,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,5,7,4,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,3,4,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,3,5,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,4,3,1,5] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,4,5,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,5,3,1,4] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
[6,2,7,5,4,1,3] => [6,2,7,5,4,1,3] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 4 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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