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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000724
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000724: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ 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 π such that π(1)<π(2)>π(3)⋯) with the greater neighbor of the maximum ([[St000060]]), see also [3].
Matching statistic: St001207
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%
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] => ? = 2
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[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]/(xn).
Matching statistic: St001582
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%
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] => ? = 2
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2
[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
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%
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] => ? = 2 + 3
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 3
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 3
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 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 Ext1A(Io,A) when Io is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(xn).
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