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Your data matches 123 different statistics following compositions of up to 3 maps.
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Matching statistic: St001198
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000782
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,-3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,-4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[2,-1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-2,1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-2,-1,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 2 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 2 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 2 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 2 - 1
[2,3,-1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,-1,-4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[2,-3,1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-3,1,-4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-2,3,1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,3,1,-4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 2 - 1
[2,3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,-3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,-3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[-2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[-2,3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[-2,-3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[-2,-3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
[2,-4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 2 - 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001771
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
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 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
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 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => 0 = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[-2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 2
[2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[2,-1,-4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,4,-3,-5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
[-2,1,-4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 2
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: St001772
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => [-1,-2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
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
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => [-1,-2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
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
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => [-1,-2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 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: St001867
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
[-1,-2] => [1,2] => [1,2] => [-1,-2] => 0 = 2 - 2
[1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 2 - 2
[-1,3,-2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[-1,-3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 2 - 2
[2,-1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[-2,1,-3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 2 - 2
[2,3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[2,-3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[-2,-3,-1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 2 - 2
[3,1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[-3,-1,-2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0 = 2 - 2
[3,-2,-1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[-3,-2,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0 = 2 - 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 2 - 2
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 2 - 2
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 2 - 2
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,3,-4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,-3,-4,-2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 2 - 2
[1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[-1,5,-4,-3,-2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 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,-5,-4,-3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 2 - 2
[2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[-2,-1,-3,-4,-5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 2 - 2
[2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[2,-1,-3,-5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,3,-5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
[-2,1,-3,5,-4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 2 - 2
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 113 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. St000068The number of minimal elements in a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001720The minimal length of a chain of small intervals in a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001768The number of reduced words of a signed permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001625The Möbius invariant of a lattice. St001730The number of times the path corresponding to a binary word crosses the base line. St001927Sparre Andersen's number of positives of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St000911The number of maximal antichains of maximal size in a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001472The permanent of the Coxeter matrix of the poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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