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Matching statistic: St000713
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> 5
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 5
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 5
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 5
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 5
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 5
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 5
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 5
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 5
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 5
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 5
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 5
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 5
[2,1,4,3] => [2,2]
=> [2]
=> 10
[2,1,-4,-3] => [2,2]
=> [2]
=> 10
[-2,-1,4,3] => [2,2]
=> [2]
=> 10
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 10
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 5
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 5
[3,4,1,2] => [2,2]
=> [2]
=> 10
[3,-4,1,-2] => [2,2]
=> [2]
=> 10
[-3,4,-1,2] => [2,2]
=> [2]
=> 10
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 10
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 5
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 5
[4,3,2,1] => [2,2]
=> [2]
=> 10
[4,-3,-2,1] => [2,2]
=> [2]
=> 10
[-4,3,2,-1] => [2,2]
=> [2]
=> 10
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 10
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> 5
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> 5
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> 5
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> 5
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> 5
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> 5
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> 5
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> 5
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> 5
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> 5
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,-4] => [1,1,1]
=> [1,1]
=> 5
[1,2,3,-5,4] => [1,1,1]
=> [1,1]
=> 5
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,1,1]
=> 0
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
Consider the symplectic group Sp(2n). Then the integer partition (μ1,…,μk) of length at most n corresponds to the weight vector (μ1−μ2,…,μk−2−μk−1,μn,0,…,0).
For example, the integer partition (2) labels the symmetric square of the vector representation, whereas the integer partition (1,1) labels the second fundamental representation.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
[1,2,3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[1,2,3,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-3,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-2,3,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-1,2,3,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,2,-4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-3,-2,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-4,3,-2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-2,-1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[2,1,4,3] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[2,1,-4,-3] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-2,-1,4,3] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-2,-1,-4,-3] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-3,2,-1,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[3,4,1,2] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[3,-4,1,-2] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-3,4,-1,2] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-3,-4,-1,-2] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-4,2,3,-1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[4,3,2,1] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[4,-3,-2,1] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-4,3,2,-1] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[-4,-3,-2,-1] => [2,2]
=> [4]
=> 10000 => ? = 10 - 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => ? = 0 - 3
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 0 - 3
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 - 3
[1,2,3,5,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,-5,4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,-5,-4] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 - 3
[1,2,-3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,2,-3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-2,3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-2,3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-1,2,3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-1,2,3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 - 3
[1,2,4,3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,2,4,-3,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-4,3,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-4,-3,5] => [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 - 3
[1,2,-4,-3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-2,4,3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,-2,-4,-3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-1,2,4,3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[-1,2,-4,-3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 5 - 3
[1,2,4,5,3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 5 - 3
[1,2,4,-5,-3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 5 - 3
[1,2,-4,5,-3] => [3,1,1]
=> [2,1,1,1]
=> 101110 => ? = 5 - 3
[1,2,5,4,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,-5,4,3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,3,-2,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-3,2,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,4,3,-2,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-4,3,2,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,5,3,4,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,-5,3,4,2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[2,-1,3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-2,1,3,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[3,2,-1,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-3,2,1,4,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[4,2,3,-1,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-4,2,3,1,5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[5,2,3,4,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-5,2,3,4,1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,6,4,-5] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,3,5,6,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,5,4,6,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,6,3,5,-4] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,2,4,6,5,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,5,3,4,6,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,4,3,6,5,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,6,2,4,5,-3] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[1,3,6,4,5,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[5,2,3,4,6,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[4,2,3,6,5,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[3,2,6,4,5,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[6,1,3,4,5,-2] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[2,6,3,4,5,-1] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
[-3,-2,1,4,5,6] => [1,1,1]
=> [1,1,1]
=> 1110 => 2 = 5 - 3
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-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 MS. 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)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 10 - 5
[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]]
=> ? = 0 - 5
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 - 5
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 0 - 5
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,-5,-4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 0 - 5
[1,2,-3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,-3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-1,2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-1,2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 0 - 5
[1,2,4,3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,4,-3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-4,3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-4,-3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 0 - 5
[1,2,-4,-3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,-2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-1,2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[-1,2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,4,-5,-3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,-4,5,-3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5 - 5
[1,2,5,4,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,-5,4,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,3,-2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-3,2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,4,3,-2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-4,3,2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,5,3,4,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,-5,3,4,2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[2,-1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-2,1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[3,2,-1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-3,2,1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[4,2,3,-1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-4,2,3,1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[5,2,3,4,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-5,2,3,4,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,6,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,3,5,6,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,5,4,6,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,6,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,2,4,6,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,5,3,4,6,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,4,3,6,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,6,2,4,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[1,3,6,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[5,2,3,4,6,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[4,2,3,6,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[3,2,6,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[6,1,3,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[2,6,3,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
[-3,-2,1,4,5,6] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 5 - 5
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
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