Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000939
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 4
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 5
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 3
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 3
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 6
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 2
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 2
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 2
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St001491
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St001491: Binary words ⟶ ℤResult quality: 12% values known / values provided: 13%distinct values known / distinct values provided: 12%
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 10110 => ? = 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 11010 => ? = 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 11100 => ? = 3
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1100 => 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 10100 => ? = 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 11000 => ? = 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> 1100 => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => ? = 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => ? = 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => ? = 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> 11100 => ? = 3
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> 10100 => ? = 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> 11000 => ? = 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 111110 => ? = 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> 101110 => ? = 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> 110110 => ? = 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> 10110 => ? = 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> 100110 => ? = 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> 111010 => ? = 4
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> 11010 => ? = 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> 101010 => ? = 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 110010 => ? = 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 111100 => ? = 5
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> 11100 => ? = 3
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 101100 => ? = 3
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1100 => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 110100 => ? = 3
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 10100 => ? = 1
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> 100100 => ? = 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 111000 => ? = 6
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 11000 => ? = 2
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> 101000 => ? = 2
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 110000 => ? = 2
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => ? = 3
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 2
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> [2,1,1]
=> 10110 => ? = 2
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
=> [2,2,1]
=> 11010 => ? = 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> [2,2]
=> 1100 => 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
=> [3,2]
=> 10100 => ? = 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 111110 => ? = 5
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => ? = 3
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]]
=> [2,1,1,1]
=> 101110 => ? = 3
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
=> [2,2,1,1]
=> 110110 => ? = 2
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => ? = 2
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
=> [3,1,1]
=> 100110 => ? = 2
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
=> [2,2,2,1]
=> 111010 => ? = 4
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => ? = 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => ? = 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]]
=> [3,3,1]
=> 110010 => ? = 2
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> 11100 => ? = 3
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]]
=> [3,2,2]
=> 101100 => ? = 3
[1,3,1,3] => [[5,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 1
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]]
=> [3,3,2]
=> 110100 => ? = 3
[1,3,2,2] => [[5,4,3,1],[3,2]]
=> [3,2]
=> 10100 => ? = 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.