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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000675
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St001491
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[2,-1] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,1] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 1 - 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 1 - 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 1 - 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 1 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 1 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 - 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 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]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-3,2,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 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]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-1,3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-1,4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3 - 1
[1,4,3,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-4,3,2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[2,-1,3,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,1,3,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[2,1,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,1,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-1,4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-1,-4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,1,4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,1,-4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-1,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-1,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,2,-1,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-2,1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-3,2,1,4] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-3,-2,-1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[3,4,1,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,4,-1,2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-4,1,2] => [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-4,-1,-2] => [2]
=> [1,0,1,0]
=> 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.
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