Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001956
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001956: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [1,0]
 => 0
[1,-2] => [1]
 => [1,0]
 => 0
[-1,2] => [1]
 => [1,0]
 => 0
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 1
[2,-1] => [2]
 => [1,0,1,0]
 => 0
[-2,1] => [2]
 => [1,0,1,0]
 => 0
[1,2,-3] => [1]
 => [1,0]
 => 0
[1,-2,3] => [1]
 => [1,0]
 => 0
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 1
[-1,2,3] => [1]
 => [1,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,3,-2] => [2]
 => [1,0,1,0]
 => 0
[1,-3,2] => [2]
 => [1,0,1,0]
 => 0
[-1,3,2] => [1]
 => [1,0]
 => 0
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-1,-3,-2] => [1]
 => [1,0]
 => 0
[2,1,-3] => [1]
 => [1,0]
 => 0
[2,-1,3] => [2]
 => [1,0,1,0]
 => 0
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-2,1,3] => [2]
 => [1,0,1,0]
 => 0
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-2,-1,-3] => [1]
 => [1,0]
 => 0
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 1
[3,2,-1] => [2]
 => [1,0,1,0]
 => 0
[3,-2,1] => [1]
 => [1,0]
 => 0
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-3,2,1] => [2]
 => [1,0,1,0]
 => 0
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[-3,-2,-1] => [1]
 => [1,0]
 => 0
[1,2,3,-4] => [1]
 => [1,0]
 => 0
[1,2,-3,4] => [1]
 => [1,0]
 => 0
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,4] => [1]
 => [1,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[-1,2,3,4] => [1]
 => [1,0]
 => 0
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
Description
The comajor index for set-valued two-row standard Young tableaux. The comajorindex is the sum $\sum_k (n+1-k)$ over all natural descents $k$. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let $j$ be smallest integer such that $2j$ is a down step. Then $k$ is a natural descent if * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a valley i.e. [0,1], or * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a peak i.e. [1,0], or * $k-1\ge j$ and positions $2(k-1),2k-1,2k$ form [0,1,1], or * $k=j$ and positions $2k-1,2k$ are double down i.e. [0,0], or * $k < j$ and positions $2k-1,2k$ are a valley i.e. [0,1].
Matching statistic: St001491
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 17%●distinct values known / distinct values provided: 5%
Values
[-1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[1,2,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-2,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,2,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[-1,3,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-1,-3,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,1,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-2,-1,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[3,-2,1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-3,-2,-1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,2,3,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,2,-3,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[1,-2,3,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3
[-1,2,3,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 6
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[1,-2,4,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[1,-2,-4,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,2,4,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[-1,2,-4,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 1
[1,3,2,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 0
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 2
[1,-3,-2,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,3,2,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-3,-2,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,3,4,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,3,-4,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-3,4,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-3,-4,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,4,2,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,4,-2,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-4,2,-3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-4,-2,3] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,4,-3,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[1,-4,-3,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,4,3,2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-1,-4,3,-2] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,1,3,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,1,-3,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,-1,3,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,-1,-3,4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,3,1,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,-3,-1,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,3,-1,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,-3,1,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,4,-3,1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[2,-4,-3,-1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,4,-3,-1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[-2,-4,-3,1] => [1]
 => [1,0,1,0]
 => 1010 => 0
[3,1,2,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
[3,-1,-2,-4] => [1]
 => [1,0,1,0]
 => 1010 => 0
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.