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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000981
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000981: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000981: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,-2,4,-3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,3,-2,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-3,2,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,4,-3,-2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-4,-3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,-1,-3,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-2,1,-3,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,-1,4,-3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[2,-1,-4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-2,1,4,-3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-2,1,-4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[3,-2,-1,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-3,-2,1,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,4,-1,-2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[3,-4,-1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-3,4,1,-2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-3,-4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[4,-2,-3,-1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-4,-2,-3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,3,-2,-1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[4,-3,2,-1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-4,3,-2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[-4,-3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 4
[1,2,-3,-4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,-4,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,3,-4,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,-2,-3,4,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,-2,-3,-4,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 6
[1,-2,-3,5,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,-2,-3,-5,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,5,-4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[-1,2,-3,-5,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
Description
The length of the longest zigzag subpath.
This is the length of the longest consecutive subpath that is a zigzag of the form $010...$ or of the form $101...$.
Matching statistic: St001582
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Values
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[3,-2,-1,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-3,-2,1,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[3,4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[3,-4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-3,4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-3,-4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[4,-2,-3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-4,-2,-3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[4,3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[4,-3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-4,3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-4,-3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,2,-3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 6 - 1
[1,-2,-3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-3,5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-3,-5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-3,-5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,4,3,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,4,-3,-5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-4,3,-5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-4,-3,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,4,5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,4,-5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-4,-5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,5,3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,5,-3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-5,-3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[1,-2,5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,5,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,5,-4,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,5,-4,-3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-5,-4,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,3,2,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-3,-2,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,3,-2,5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,3,-2,-5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-3,2,5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-3,2,-5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-1,4,-3,2,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-4,-3,-2,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,4,5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,4,-5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-4,5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-4,-5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-1,5,-3,-4,2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-5,-3,-4,-2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,5,4,-3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,5,-4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-5,4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001583
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Values
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[3,-2,-1,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-3,-2,1,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[3,4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[3,-4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-3,4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-3,-4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[4,-2,-3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-4,-2,-3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[4,3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[4,-3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-4,3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-4,-3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,2,-3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,2,-3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,3,-4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 4 - 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 6 - 1
[1,-2,-3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,3,5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,3,-5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-3,5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-3,-5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-3,-5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,-2,4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,4,3,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,4,-3,-5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-4,3,-5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-4,-3,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,4,5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,4,-5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-4,-5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,5,3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,5,-3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[-1,-2,-5,-3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 - 1
[1,-2,5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[1,-2,-5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,2,-5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,5,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,5,-4,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-2,5,-4,-3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 4 - 1
[-1,-2,-5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 - 1
[-1,-2,-5,-4,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,3,2,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-3,-2,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,3,-2,5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,3,-2,-5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-3,2,5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-3,2,-5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-1,4,-3,2,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-4,-3,-2,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,4,5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,4,-5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-4,5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-4,-5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[-1,5,-3,-4,2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[-1,-5,-3,-4,-2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[1,5,4,-3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,5,-4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,-5,4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001603
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[-1,-2,-3] => [-1,-2,-3] => []
=> ?
=> ? = 2 - 3
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 4 - 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,4,-3,-2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-4,-3,2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
=> []
=> ? = 2 - 3
[-2,1,-3,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 3
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 3
[-2,1,4,-3] => [-2,-1,-3,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-2,1,-4,3] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 3
[3,-2,-1,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 2 - 3
[-3,-2,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[3,4,-1,-2] => [-1,-2,3,4] => [1,1]
=> [1]
=> ? = 4 - 3
[3,-4,-1,2] => [-1,-2,3,-4] => [1]
=> []
=> ? = 4 - 3
[-3,4,1,-2] => [-3,-2,-1,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-3,-4,1,2] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 3
[4,-2,-3,-1] => [-1,-2,-3,4] => [1]
=> []
=> ? = 2 - 3
[-4,-2,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[4,3,-2,-1] => [-1,-2,4,3] => [2]
=> []
=> ? = 4 - 3
[4,-3,2,-1] => [-1,-3,-2,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-4,3,-2,1] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 3
[-4,-3,2,1] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 3
[1,2,-3,-4,-5] => [1,2,-3,-4,-5] => [1,1]
=> [1]
=> ? = 2 - 3
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 4 - 3
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 6 - 3
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,5,-4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-5,4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[1,-3,2,-5,4] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-4,-5,2,3] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-5,4,-3,2] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-5,-4,3,2] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,3,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-4,3,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,4,-5,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-4,5,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-5,3,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-5,4,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-3,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,3,1,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-4,-5,1,3] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,4,-5,1,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-5,4,-3,1] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-5,-4,3,1] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,5,-4,3,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,1,2,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,2,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,1,2,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,4,1,-5,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,1,5,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,-4,-5,2,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,5,2,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,1,2,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,1,4,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,-5,-4,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,4,1,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,1,-5,2,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,2,-5,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-2,1,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,2,1,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,-3,2,-5,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-2,5,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,2,5,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-5,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,5,1,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,-5,1,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,1,3,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,5,-2,1,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,2,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,3,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,1,4,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,1,-4,3,2] => [-5,-4,3,-2,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,2,4,-3,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,2,-4,3,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,3,-2,1,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,-3,2,1,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,3,-2,4,1] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,-3,2,4,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[-1,-2,-3] => [-1,-2,-3] => []
=> ?
=> ? = 2 - 3
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 4 - 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,4,-3,-2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[-1,-4,-3,2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 3
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
=> []
=> ? = 2 - 3
[-2,1,-3,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 3
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 3
[-2,1,4,-3] => [-2,-1,-3,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-2,1,-4,3] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 3
[3,-2,-1,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 2 - 3
[-3,-2,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[3,4,-1,-2] => [-1,-2,3,4] => [1,1]
=> [1]
=> ? = 4 - 3
[3,-4,-1,2] => [-1,-2,3,-4] => [1]
=> []
=> ? = 4 - 3
[-3,4,1,-2] => [-3,-2,-1,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-3,-4,1,2] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 3
[4,-2,-3,-1] => [-1,-2,-3,4] => [1]
=> []
=> ? = 2 - 3
[-4,-2,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 3
[4,3,-2,-1] => [-1,-2,4,3] => [2]
=> []
=> ? = 4 - 3
[4,-3,2,-1] => [-1,-3,-2,4] => [2,1]
=> [1]
=> ? = 4 - 3
[-4,3,-2,1] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 3
[-4,-3,2,1] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 3
[1,2,-3,-4,-5] => [1,2,-3,-4,-5] => [1,1]
=> [1]
=> ? = 2 - 3
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 4 - 3
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,-2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 3
[-1,-2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 6 - 3
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 3
[-1,2,-3,5,-4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[-1,2,-3,-5,4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 3
[1,-3,2,-5,4] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-4,-5,2,3] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-5,4,-3,2] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[1,-5,-4,3,2] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,3,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-4,3,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,4,-5,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-4,5,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-5,3,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,1,-5,4,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-3,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,3,1,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-4,-5,1,3] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,4,-5,1,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-5,4,-3,1] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,-5,-4,3,1] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-2,5,-4,3,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,1,2,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,2,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,1,2,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,4,1,-5,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,1,5,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,-4,-5,2,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-4,5,2,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,1,2,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,1,4,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,-5,-4,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-3,-5,4,1,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,1,-5,2,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,2,-5,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-2,1,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,2,1,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,-3,2,-5,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-2,5,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,2,5,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,3,-5,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-3,5,1,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,-5,1,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,1,3,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,5,-2,1,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,2,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-4,-5,3,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,1,4,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,1,-4,3,2] => [-5,-4,3,-2,-1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,2,4,-3,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,2,-4,3,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,3,-2,1,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,-3,2,1,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,3,-2,4,1] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[-5,-3,2,4,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[-1,-2,-3] => [-1,-2,-3] => []
=> ?
=> ? = 2 - 4
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 2 - 4
[-1,2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-2,3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-2,-3,4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-2,-3,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 4 - 4
[-1,-2,4,-3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-2,-4,3] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,3,-2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-3,2,-4] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,4,-3,-2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[-1,-4,-3,2] => [-1,-2,-3,-4] => []
=> ?
=> ? = 2 - 4
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
=> []
=> ? = 2 - 4
[-2,1,-3,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 4
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 4
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
=> []
=> ? = 4 - 4
[-2,1,4,-3] => [-2,-1,-3,4] => [2,1]
=> [1]
=> ? = 4 - 4
[-2,1,-4,3] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 4
[3,-2,-1,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 2 - 4
[-3,-2,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 4
[3,4,-1,-2] => [-1,-2,3,4] => [1,1]
=> [1]
=> ? = 4 - 4
[3,-4,-1,2] => [-1,-2,3,-4] => [1]
=> []
=> ? = 4 - 4
[-3,4,1,-2] => [-3,-2,-1,4] => [2,1]
=> [1]
=> ? = 4 - 4
[-3,-4,1,2] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 4
[4,-2,-3,-1] => [-1,-2,-3,4] => [1]
=> []
=> ? = 2 - 4
[-4,-2,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 2 - 4
[4,3,-2,-1] => [-1,-2,4,3] => [2]
=> []
=> ? = 4 - 4
[4,-3,2,-1] => [-1,-3,-2,4] => [2,1]
=> [1]
=> ? = 4 - 4
[-4,3,-2,1] => [-2,-1,-4,-3] => [2,2]
=> [2]
=> ? = 4 - 4
[-4,-3,2,1] => [-3,-4,-1,-2] => [2,2]
=> [2]
=> ? = 4 - 4
[1,2,-3,-4,-5] => [1,2,-3,-4,-5] => [1,1]
=> [1]
=> ? = 2 - 4
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 4 - 4
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[-1,2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 4
[-1,-2,3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,-2,3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,-2,3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 4
[-1,-2,-3,4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,-2,-3,4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 4
[-1,-2,-3,-4,5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 4 - 4
[-1,-2,-3,-4,-5] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 6 - 4
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1]
=> []
=> ? = 2 - 4
[-1,2,-3,5,-4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[-1,2,-3,-5,4] => [-1,-2,-3,-4,-5] => []
=> ?
=> ? = 2 - 4
[1,-3,2,-5,4] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[1,-4,-5,2,3] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[1,-5,4,-3,2] => [1,-3,-2,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[1,-5,-4,3,2] => [1,-4,-5,-2,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,3,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,-4,3,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,4,-5,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,-4,5,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,-5,3,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,1,-5,4,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[2,-3,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,3,1,-5,4] => [-2,-1,3,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[2,-4,-5,1,3] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,4,-5,1,3] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[2,-5,4,-3,1] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[2,-5,-4,3,1] => [-4,2,-5,-1,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-2,5,-4,3,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,1,2,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,2,1,-5,4] => [-3,2,-1,-5,-4] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-4,1,2,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,4,1,-5,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-4,1,5,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[3,-4,-5,2,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-4,5,2,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-5,1,2,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-5,1,4,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[3,-5,-4,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-3,-5,4,1,2] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,1,-5,2,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,2,-5,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,3,-2,1,5] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-3,2,1,5] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[4,-3,2,-5,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,3,-2,5,1] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-3,2,5,1] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,3,-5,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-3,5,1,2] => [-3,-4,-1,-2,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[4,-5,1,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-5,1,3,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,5,-2,1,3] => [-2,-1,-4,-3,5] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-5,2,1,3] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-4,-5,3,1,2] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,1,4,-3,2] => [-5,-3,-2,4,-1] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,1,-4,3,2] => [-5,-4,3,-2,-1] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,2,4,-3,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,2,-4,3,1] => [-4,-5,3,-1,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,3,-2,1,4] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,-3,2,1,4] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,3,-2,4,1] => [-2,-1,-5,4,-3] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
[-5,-3,2,4,1] => [-3,-5,-1,4,-2] => [2,2,1]
=> [2,1]
=> 0 = 4 - 4
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Values
[-1,-2,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,-2,4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,-4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,3,-2,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-3,2,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,4,-3,-2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-4,-3,2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[2,-1,-3,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-2,1,-3,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[2,-1,4,-3] => [2,2]
=> [4]
=> 10000 => ? = 4
[2,-1,-4,3] => [2,2]
=> [4]
=> 10000 => ? = 4
[-2,1,4,-3] => [2,2]
=> [4]
=> 10000 => ? = 4
[-2,1,-4,3] => [2,2]
=> [4]
=> 10000 => ? = 4
[3,-2,-1,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-3,-2,1,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[3,4,-1,-2] => [2,2]
=> [4]
=> 10000 => ? = 4
[3,-4,-1,2] => [2,2]
=> [4]
=> 10000 => ? = 4
[-3,4,1,-2] => [2,2]
=> [4]
=> 10000 => ? = 4
[-3,-4,1,2] => [2,2]
=> [4]
=> 10000 => ? = 4
[4,-2,-3,-1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-4,-2,-3,1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[4,3,-2,-1] => [2,2]
=> [4]
=> 10000 => ? = 4
[4,-3,2,-1] => [2,2]
=> [4]
=> 10000 => ? = 4
[-4,3,-2,1] => [2,2]
=> [4]
=> 10000 => ? = 4
[-4,-3,2,1] => [2,2]
=> [4]
=> 10000 => ? = 4
[1,2,-3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,-3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,-3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,2,3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,-3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,-3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,-3,-4,-5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,-2,3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,3,-4,-5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,-2,-3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,-3,4,-5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,-2,-3,-4,5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 4
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => ? = 6
[1,-2,-3,5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[1,-2,-3,-5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,2,-3,5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,2,-3,-5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,3,5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,3,-5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,-3,5,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 4
[-1,-2,-3,-5,4] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 4
[-1,-2,-3,-5,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,4,-3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[1,-2,-4,3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,2,4,-3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,2,-4,3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,4,3,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,4,-3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,4,-3,-5] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 4
[-1,-2,-4,3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 2
[-1,-2,-4,3,-5] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 4
[-1,-2,-4,-3,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,4,5,-3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 2
[-1,-2,4,-5,3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 2
[-1,-2,-4,5,3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 2
[-1,-2,5,-4,3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-2,-5,-4,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,3,2,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-3,-2,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,4,-3,2,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-4,-3,-2,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,5,-3,-4,2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-1,-5,-3,-4,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[2,1,-3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-2,-1,-3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[3,-2,1,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-3,-2,-1,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[4,-2,-3,1,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-4,-2,-3,-1,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[5,-2,-3,-4,1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[-5,-2,-3,-4,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[7,-2,1,-4,3,-6,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
[6,-2,3,-4,-5,1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001713
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Values
[-1,-2,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,-4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,3,-2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-3,2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,4,-3,-2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-4,-3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[2,-1,-3,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-2,1,-3,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[2,-1,4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[2,-1,-4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-2,1,4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-2,1,-4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[3,-2,-1,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-3,-2,1,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[3,4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[3,-4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-3,4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-3,-4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[4,-2,-3,-1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-4,-2,-3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[4,3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[4,-3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-4,3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[-4,-3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 - 2
[1,2,-3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,2,-3,-4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,3,-4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,-3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 6 - 2
[1,-2,-3,5,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[1,-2,-3,-5,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,2,-3,5,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,2,-3,-5,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,3,5,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,3,-5,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,-3,5,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-3,-5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-3,-5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[1,-2,4,-3,-5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[1,-2,-4,3,-5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,2,4,-3,-5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,2,-4,3,-5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,4,3,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,4,-3,5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,4,-3,-5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-4,3,5] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,-4,3,-5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 - 2
[-1,-2,-4,-3,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,4,5,-3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,4,-5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,-4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 - 2
[-1,-2,5,-4,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-2,-5,-4,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,3,2,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-3,-2,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,4,-3,2,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-4,-3,-2,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,5,-3,-4,2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-1,-5,-3,-4,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[2,1,-3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-2,-1,-3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[3,-2,1,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-3,-2,-1,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[4,-2,-3,1,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-4,-2,-3,-1,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[5,-2,-3,-4,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[-5,-2,-3,-4,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[7,-2,1,-4,3,-6,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
[6,-2,3,-4,-5,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 2 - 2
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
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