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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000939
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,-2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,-2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,-3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,-3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,-3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[-1,3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,-3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,-3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,-4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[1,-4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[-1,4,2,3] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,-4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[-1,-4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,-4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,4,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[-1,-4,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
Description
The number of characters of the symmetric group whose value on the partition is positive.
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
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 25%●distinct values known / distinct values provided: 14%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 25%●distinct values known / distinct values provided: 14%
Values
[1,3,2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,-3,-2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[2,1,3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-2,-1,3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[2,3,1] => [3]
=> [3]
=> 1000 => 1
[2,-3,-1] => [3]
=> [3]
=> 1000 => 1
[-2,3,-1] => [3]
=> [3]
=> 1000 => 1
[-2,-3,1] => [3]
=> [3]
=> 1000 => 1
[3,1,2] => [3]
=> [3]
=> 1000 => 1
[3,-1,-2] => [3]
=> [3]
=> 1000 => 1
[-3,1,-2] => [3]
=> [3]
=> 1000 => 1
[-3,-1,2] => [3]
=> [3]
=> 1000 => 1
[3,2,1] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-3,2,-1] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,2,3,4] => [1,1,1,1]
=> [2,2]
=> 1100 => 1
[1,2,4,3] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,2,-4,-3] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,-2,4,3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,-2,-4,-3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,4,3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,2,-4,-3] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,3,2,4] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,3,2,-4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,-3,-2,4] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,-3,-2,-4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,3,2,4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,-3,-2,4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,3,4,2] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,3,-4,-2] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,-3,4,-2] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,-3,-4,2] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-1,3,4,2] => [3]
=> [3]
=> 1000 => 1
[-1,3,-4,-2] => [3]
=> [3]
=> 1000 => 1
[-1,-3,4,-2] => [3]
=> [3]
=> 1000 => 1
[-1,-3,-4,2] => [3]
=> [3]
=> 1000 => 1
[1,4,2,3] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,4,-2,-3] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,-4,2,-3] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[1,-4,-2,3] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-1,4,2,3] => [3]
=> [3]
=> 1000 => 1
[-1,4,-2,-3] => [3]
=> [3]
=> 1000 => 1
[-1,-4,2,-3] => [3]
=> [3]
=> 1000 => 1
[-1,-4,-2,3] => [3]
=> [3]
=> 1000 => 1
[1,4,3,2] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,4,-3,2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[1,-4,3,-2] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,-4,-3,-2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,4,3,2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-1,-4,3,-2] => [2,1]
=> [1,1,1]
=> 1110 => 2
[2,1,3,4] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[2,1,3,-4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[2,1,-3,4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-2,-1,3,4] => [2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[-2,-1,3,-4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[-2,-1,-3,4] => [2,1]
=> [1,1,1]
=> 1110 => 2
[2,3,1,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[2,3,1,-4] => [3]
=> [3]
=> 1000 => 1
[2,-3,-1,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[2,-3,-1,-4] => [3]
=> [3]
=> 1000 => 1
[-2,3,-1,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-2,3,-1,-4] => [3]
=> [3]
=> 1000 => 1
[-2,-3,1,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-2,-3,1,-4] => [3]
=> [3]
=> 1000 => 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,3,-4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-3,4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-3,-4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,3,4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,3,-4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-3,-4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,-1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-4,1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-4,-1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,4,1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,4,-1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-4,-1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,3,1] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[2,4,-3,1] => [3]
=> [3]
=> 1000 => 1
[2,-4,3,-1] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[2,-4,-3,-1] => [3]
=> [3]
=> 1000 => 1
[-2,4,3,-1] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-2,4,-3,-1] => [3]
=> [3]
=> 1000 => 1
[-2,-4,3,1] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-2,-4,-3,1] => [3]
=> [3]
=> 1000 => 1
[3,1,2,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[3,1,2,-4] => [3]
=> [3]
=> 1000 => 1
[3,-1,-2,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[3,-1,-2,-4] => [3]
=> [3]
=> 1000 => 1
[-3,1,-2,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[-3,1,-2,-4] => [3]
=> [3]
=> 1000 => 1
[-3,-1,2,4] => [3,1]
=> [3,1]
=> 10010 => ? = 4
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,1,-4,-2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,-1,4,-2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,-1,-4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-3,1,4,-2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-3,1,-4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
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: St000075
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> ? = 1 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,-2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,-3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,-3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-1,3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,-3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,-3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,-4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[1,-4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-1,4,2,3] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,-4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-1,-4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[1,-4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,4,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[2,1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 + 1
[-2,-1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[2,3,1,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[2,-3,-1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[2,-3,-1,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,3,-1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-2,3,-1,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,-3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-2,-3,1,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,-3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,-3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,-3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,-3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,-4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,-4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,-4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-2,-4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[2,4,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[2,-4,3,-1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[2,-4,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,4,3,-1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-2,4,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-2,-4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-2,-4,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[3,1,2,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,-1,-2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[3,-1,-2,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-3,1,-2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-3,1,-2,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[-3,-1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 4 + 1
[-3,-1,2,-4] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[3,1,-4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[3,-1,4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[3,-1,-4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
[-3,1,4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 3 + 1
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St001168
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001168: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001168: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 23
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,-2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,-2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,2,4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,2,-4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,-3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,-3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,-3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-1,3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,-3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,-3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,-4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[1,-4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-1,4,2,3] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,-4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-1,-4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[1,-4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,4,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-1,-4,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[2,1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[2,1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 23
[-2,-1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[-2,-1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 25 = 2 + 23
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[2,3,1,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[2,-3,-1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[2,-3,-1,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,3,-1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-2,3,-1,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,-3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-2,-3,1,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,-3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,-3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,-3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,-3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,-4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,-4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,-4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-2,-4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[2,4,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[2,-4,3,-1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[2,-4,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,4,3,-1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-2,4,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-2,-4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-2,-4,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[3,1,2,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,-1,-2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[3,-1,-2,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-3,1,-2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-3,1,-2,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[-3,-1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 4 + 23
[-3,-1,2,-4] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 24 = 1 + 23
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[3,1,-4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[3,-1,4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[3,-1,-4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
[-3,1,4,-2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 23
Description
The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000782
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 - 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,-3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[1,-4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2 - 1
[-2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 3 - 1
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001207
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 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,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001582
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 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,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 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
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 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,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 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: St001722
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1 - 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,-3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[1,-4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[-2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001816
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 12%●distinct values known / distinct values provided: 7%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,-1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1 - 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,-2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,-3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,-3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,-3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[-1,3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,-3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,-3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,-4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[1,-4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[-1,4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,-4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-1,-4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,-4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[-2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[2,3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[2,-3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[2,-3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-2,3,-1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[-2,3,-1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[-2,-3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 4 - 2
[-2,-3,1,-4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 1 - 2
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[-2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[-2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[-2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[-2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3 - 2
[3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[4,-2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,3,5,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,3,-5,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,4,3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-4,-3,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,4,3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,-4,-3,5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,5,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-5,-4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,5,4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,-5,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,3,2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-3,-2,-4,-5] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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