- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St001491: Binary words ⟶ ℤ

Values

[[2,1],[1]] => [1] => 10 => 11 => 2

[[3,1],[1]] => [1] => 10 => 11 => 2

[[2,2],[1]] => [1] => 10 => 11 => 2

[[3,2],[2]] => [2] => 100 => 011 => 1

[[2,2,1],[1,1]] => [1,1] => 110 => 111 => 3

[[2,1,1],[1]] => [1] => 10 => 11 => 2

[[3,2,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,1],[1]] => [1] => 10 => 11 => 2

[[3,2],[1]] => [1] => 10 => 11 => 2

[[4,2],[2]] => [2] => 100 => 011 => 1

[[3,2,1],[1,1]] => [1,1] => 110 => 111 => 3

[[3,1,1],[1]] => [1] => 10 => 11 => 2

[[4,2,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,3],[2]] => [2] => 100 => 011 => 1

[[4,3],[3]] => [3] => 1000 => 0011 => 1

[[2,2,1],[1]] => [1] => 10 => 11 => 2

[[3,3,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,2,1],[2]] => [2] => 100 => 011 => 1

[[2,2,2],[1,1]] => [1,1] => 110 => 111 => 3

[[3,3,2],[2,2]] => [2,2] => 1100 => 0111 => 2

[[3,2,2],[2,1]] => [2,1] => 1010 => 1101 => 2

[[2,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[2,2,1,1],[1,1]] => [1,1] => 110 => 111 => 3

[[2,1,1,1],[1]] => [1] => 10 => 11 => 2

[[3,2,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[5,1],[1]] => [1] => 10 => 11 => 2

[[4,2],[1]] => [1] => 10 => 11 => 2

[[5,2],[2]] => [2] => 100 => 011 => 1

[[4,2,1],[1,1]] => [1,1] => 110 => 111 => 3

[[4,1,1],[1]] => [1] => 10 => 11 => 2

[[5,2,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,3],[1]] => [1] => 10 => 11 => 2

[[4,3],[2]] => [2] => 100 => 011 => 1

[[5,3],[3]] => [3] => 1000 => 0011 => 1

[[3,3,1],[1,1]] => [1,1] => 110 => 111 => 3

[[3,2,1],[1]] => [1] => 10 => 11 => 2

[[4,3,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,2,1],[2]] => [2] => 100 => 011 => 1

[[3,2,2],[1,1]] => [1,1] => 110 => 111 => 3

[[4,3,2],[2,2]] => [2,2] => 1100 => 0111 => 2

[[4,2,2],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[3,2,1,1],[1,1]] => [1,1] => 110 => 111 => 3

[[3,1,1,1],[1]] => [1] => 10 => 11 => 2

[[4,2,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,4],[3]] => [3] => 1000 => 0011 => 1

[[3,3,1],[2]] => [2] => 100 => 011 => 1

[[4,3,1],[3]] => [3] => 1000 => 0011 => 1

[[2,2,2],[1]] => [1] => 10 => 11 => 2

[[3,3,2],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,2,2],[2]] => [2] => 100 => 011 => 1

[[2,2,1,1],[1]] => [1] => 10 => 11 => 2

[[3,3,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[3,2,1,1],[2]] => [2] => 100 => 011 => 1

[[3,3,3],[2,2]] => [2,2] => 1100 => 0111 => 2

[[2,2,2,1],[1,1]] => [1,1] => 110 => 111 => 3

[[3,3,2,1],[2,2]] => [2,2] => 1100 => 0111 => 2

[[3,2,2,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[2,2,2,2],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[2,2,2,1,1],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[2,2,1,1,1],[1,1]] => [1,1] => 110 => 111 => 3

[[2,1,1,1,1],[1]] => [1] => 10 => 11 => 2

[[3,2,1,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[6,1],[1]] => [1] => 10 => 11 => 2

[[5,2],[1]] => [1] => 10 => 11 => 2

[[6,2],[2]] => [2] => 100 => 011 => 1

[[5,2,1],[1,1]] => [1,1] => 110 => 111 => 3

[[5,1,1],[1]] => [1] => 10 => 11 => 2

[[6,2,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,3],[1]] => [1] => 10 => 11 => 2

[[5,3],[2]] => [2] => 100 => 011 => 1

[[6,3],[3]] => [3] => 1000 => 0011 => 1

[[4,3,1],[1,1]] => [1,1] => 110 => 111 => 3

[[4,2,1],[1]] => [1] => 10 => 11 => 2

[[5,3,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[5,2,1],[2]] => [2] => 100 => 011 => 1

[[4,2,2],[1,1]] => [1,1] => 110 => 111 => 3

[[5,3,2],[2,2]] => [2,2] => 1100 => 0111 => 2

[[5,2,2],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[4,2,1,1],[1,1]] => [1,1] => 110 => 111 => 3

[[4,1,1,1],[1]] => [1] => 10 => 11 => 2

[[5,2,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,4],[2]] => [2] => 100 => 011 => 1

[[5,4],[3]] => [3] => 1000 => 0011 => 1

[[3,3,1],[1]] => [1] => 10 => 11 => 2

[[4,4,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,3,1],[2]] => [2] => 100 => 011 => 1

[[5,3,1],[3]] => [3] => 1000 => 0011 => 1

[[3,3,2],[1,1]] => [1,1] => 110 => 111 => 3

[[4,4,2],[2,2]] => [2,2] => 1100 => 0111 => 2

[[3,2,2],[1]] => [1] => 10 => 11 => 2

[[4,3,2],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,2,2],[2]] => [2] => 100 => 011 => 1

[[3,3,2,1],[1,1,1]] => [1,1,1] => 1110 => 1111 => 4

[[3,3,1,1],[1,1]] => [1,1] => 110 => 111 => 3

[[3,2,1,1],[1]] => [1] => 10 => 11 => 2

[[4,3,1,1],[2,1]] => [2,1] => 1010 => 1101 => 2

[[4,2,1,1],[2]] => [2] => 100 => 011 => 1

[[4,3,3],[2,2]] => [2,2] => 1100 => 0111 => 2

[[3,2,2,1],[1,1]] => [1,1] => 110 => 111 => 3

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search for individual values

searching the database for the individual values of this statistic

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.

Map

path rowmotion

Description

Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Map

inner shape

Description

The inner shape of a skew partition.