Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000724
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000724: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[2,-1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-2,1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].
Matching statistic: St001207
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001582
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001171
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4 + 3
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3 + 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3 + 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3 + 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 3 + 3
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
Description
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St000782
Mapping
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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[2,-1] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[-2,1] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 3 - 2
[1,3,-2] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[2,-1,3] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-2,1,3] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-3,2,1] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 3 - 2
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 3 - 2
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 3 - 2
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 3 - 2
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => ? = 4 - 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[1,-3,2,4] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,3,2,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-3,-2,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[1,3,4,-2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,3,-4,2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,-3,4,2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,-3,-4,-2] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-1,3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[1,4,2,-3] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,4,-2,3] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,-4,2,3] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[1,-4,-2,-3] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-1,4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 4 - 2
[1,4,3,-2] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[1,-4,3,2] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,4,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,4,-3,2] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-4,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-1,-4,-3,-2] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[2,1,-3,-4] => [1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 2 - 2
[2,-1,3,4] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[2,-1,3,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[2,-1,-3,4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[-2,1,3,4] => [2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 2 - 2
[-2,1,3,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-2,1,-3,4] => [2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 3 - 2
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 4 - 2
[2,-1,4,-3] => [2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1 = 3 - 2
[2,-1,-4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1 = 3 - 2
[-2,1,4,-3] => [2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1 = 3 - 2
[-2,1,-4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1 = 3 - 2
[2,3,-1,4] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[2,-3,1,4] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
[-2,3,1,4] => [3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 3 - 2
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]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$