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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000813
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: 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
St000813: 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]
=> 3
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 3
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 3
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 3
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 3
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 3
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 3
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 3
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 3
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 3
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 3
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 3
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 3
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 3
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 47
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> 3
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> 3
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> 3
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> 3
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> 3
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> 3
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 10
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> 3
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> 3
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> 3
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 10
[1,2,3,5,-4] => [1,1,1]
=> [1,1]
=> 3
[1,2,3,-5,4] => [1,1,1]
=> [1,1]
=> 3
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,1,1]
=> 10
Description
The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 15% ●values known / values provided: 18%●distinct values known / distinct values provided: 15%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 15% ●values known / values provided: 18%●distinct values known / distinct values provided: 15%
Values
[1,2,3] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[1,2,3,-4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-3,4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-2,3,4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,2,3,4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,4,3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,-4,-3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,3,2,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-3,-2,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,4,3,2] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-4,3,-2] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[2,1,3,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-2,-1,3,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[2,1,4,3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[2,1,-4,-3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-2,-1,4,3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-2,-1,-4,-3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[3,2,1,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-3,2,-1,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[3,4,1,2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[3,-4,1,-2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-3,4,-1,2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-3,-4,-1,-2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-4,2,3,-1] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[4,3,2,1] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[4,-3,-2,1] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-4,3,2,-1] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-4,-3,-2,-1] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 111110 => ? = 47 - 1
[1,2,3,4,-5] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[1,2,3,-4,5] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[1,2,3,-4,-5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-3,4,5] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[1,2,-3,4,-5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-3,-4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-2,3,4,5] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[1,-2,3,4,-5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-2,3,-4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-2,-3,4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,2,3,4,5] => [1,1,1,1]
=> 11110 => 11110 => ? = 10 - 1
[-1,2,3,4,-5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,2,3,-4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,2,-3,4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,-2,3,4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,3,5,-4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,3,-5,4] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,3,-5,-4] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,-3,5,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,-3,-5,-4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,3,5,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,3,-5,-4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-1,2,3,5,4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-1,2,3,-5,-4] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,4,3,-5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,4,-3,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-4,3,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-4,-3,5] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,-4,-3,-5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,4,3,5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,-4,-3,5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-1,2,4,3,5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-1,2,-4,-3,5] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,4,-5,-3] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,-4,5,-3] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,-4,-5,3] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,5,-3,-4] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,-5,3,-4] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,-5,-3,4] => [3,1,1]
=> 100110 => 001110 => ? = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,5,4,-3] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,5,-4,3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,2,-5,4,3] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,2,-5,4,-3] => [2,1,1,1]
=> 101110 => 011110 => ? = 10 - 1
[1,2,-5,-4,-3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,5,4,3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,-2,-5,4,-3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[-1,2,5,4,3] => [2,1,1]
=> 10110 => 01110 => ? = 3 - 1
[1,3,-2,4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-3,2,4,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,3,2,5,4] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,3,2,-5,-4] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,-3,-2,5,4] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,-3,-2,-5,-4] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[1,4,3,-2,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-4,3,2,5] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,4,5,2,3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,4,-5,2,-3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,-4,5,-2,3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,-4,-5,-2,-3] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[1,5,3,4,-2] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[1,-5,3,4,2] => [1,1,1]
=> 1110 => 1110 => 2 = 3 - 1
[-1,5,4,3,2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,5,-4,-3,2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
[-1,-5,4,3,-2] => [2,2]
=> 1100 => 1010 => 0 = 1 - 1
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)
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: 8%
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: 8%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 3
[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]]
=> ? = 47 - 3
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 10 - 3
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[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]]
=> ? = 10 - 3
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[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]]
=> ? = 10 - 3
[1,2,-3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,2,-3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-1,2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-1,2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[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]]
=> ? = 10 - 3
[1,2,4,3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,2,4,-3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,-4,3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[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]]
=> ? = 10 - 3
[1,2,-4,-3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[1,-2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-1,2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[-1,2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3 - 3
[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]]
=> ? = 3 - 3
[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]]
=> ? = 3 - 3
[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]]
=> ? = 3 - 3
[1,2,5,4,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,-5,4,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,3,-2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-3,2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,4,3,-2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-4,3,2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,5,3,4,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,-5,3,4,2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[2,-1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-2,1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[3,2,-1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-3,2,1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[4,2,3,-1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-4,2,3,1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[5,2,3,4,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-5,2,3,4,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,3,6,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,3,5,6,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,5,4,6,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,6,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,2,4,6,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,5,3,4,6,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,4,3,6,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,6,2,4,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[1,3,6,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[5,2,3,4,6,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[4,2,3,6,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[3,2,6,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[6,1,3,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[2,6,3,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
[-3,-2,1,4,5,6] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 3 - 3
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
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