- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤ

Values

[[1,2]] => [1,1] => 110 => 010 => 1

[[1],[2]] => [1,1] => 110 => 010 => 1

[[1,3]] => [1,1] => 110 => 010 => 1

[[2,3]] => [1,1] => 110 => 010 => 1

[[1],[3]] => [1,1] => 110 => 010 => 1

[[2],[3]] => [1,1] => 110 => 010 => 1

[[1,1,2]] => [2,1] => 1010 => 0010 => 1

[[1,2,2]] => [2,1] => 1010 => 0010 => 1

[[1,1],[2]] => [2,1] => 1010 => 0010 => 1

[[1,2],[2]] => [2,1] => 1010 => 0010 => 1

[[1,4]] => [1,1] => 110 => 010 => 1

[[2,4]] => [1,1] => 110 => 010 => 1

[[3,4]] => [1,1] => 110 => 010 => 1

[[1],[4]] => [1,1] => 110 => 010 => 1

[[2],[4]] => [1,1] => 110 => 010 => 1

[[3],[4]] => [1,1] => 110 => 010 => 1

[[1,1,3]] => [2,1] => 1010 => 0010 => 1

[[1,2,3]] => [1,1,1] => 1110 => 0110 => 2

[[1,3,3]] => [2,1] => 1010 => 0010 => 1

[[2,2,3]] => [2,1] => 1010 => 0010 => 1

[[2,3,3]] => [2,1] => 1010 => 0010 => 1

[[1,1],[3]] => [2,1] => 1010 => 0010 => 1

[[1,2],[3]] => [1,1,1] => 1110 => 0110 => 2

[[1,3],[2]] => [1,1,1] => 1110 => 0110 => 2

[[1,3],[3]] => [2,1] => 1010 => 0010 => 1

[[2,2],[3]] => [2,1] => 1010 => 0010 => 1

[[2,3],[3]] => [2,1] => 1010 => 0010 => 1

[[1],[2],[3]] => [1,1,1] => 1110 => 0110 => 2

[[1,1,2,2]] => [2,2] => 1100 => 0100 => 1

[[1,1,2],[2]] => [2,2] => 1100 => 0100 => 1

[[1,1],[2,2]] => [2,2] => 1100 => 0100 => 1

[[1,5]] => [1,1] => 110 => 010 => 1

[[2,5]] => [1,1] => 110 => 010 => 1

[[3,5]] => [1,1] => 110 => 010 => 1

[[4,5]] => [1,1] => 110 => 010 => 1

[[1],[5]] => [1,1] => 110 => 010 => 1

[[2],[5]] => [1,1] => 110 => 010 => 1

[[3],[5]] => [1,1] => 110 => 010 => 1

[[4],[5]] => [1,1] => 110 => 010 => 1

[[1,1,4]] => [2,1] => 1010 => 0010 => 1

[[1,2,4]] => [1,1,1] => 1110 => 0110 => 2

[[1,3,4]] => [1,1,1] => 1110 => 0110 => 2

[[1,4,4]] => [2,1] => 1010 => 0010 => 1

[[2,2,4]] => [2,1] => 1010 => 0010 => 1

[[2,3,4]] => [1,1,1] => 1110 => 0110 => 2

[[2,4,4]] => [2,1] => 1010 => 0010 => 1

[[3,3,4]] => [2,1] => 1010 => 0010 => 1

[[3,4,4]] => [2,1] => 1010 => 0010 => 1

[[1,1],[4]] => [2,1] => 1010 => 0010 => 1

[[1,2],[4]] => [1,1,1] => 1110 => 0110 => 2

[[1,4],[2]] => [1,1,1] => 1110 => 0110 => 2

[[1,3],[4]] => [1,1,1] => 1110 => 0110 => 2

[[1,4],[3]] => [1,1,1] => 1110 => 0110 => 2

[[1,4],[4]] => [2,1] => 1010 => 0010 => 1

[[2,2],[4]] => [2,1] => 1010 => 0010 => 1

[[2,3],[4]] => [1,1,1] => 1110 => 0110 => 2

[[2,4],[3]] => [1,1,1] => 1110 => 0110 => 2

[[2,4],[4]] => [2,1] => 1010 => 0010 => 1

[[3,3],[4]] => [2,1] => 1010 => 0010 => 1

[[3,4],[4]] => [2,1] => 1010 => 0010 => 1

[[1],[2],[4]] => [1,1,1] => 1110 => 0110 => 2

[[1],[3],[4]] => [1,1,1] => 1110 => 0110 => 2

[[2],[3],[4]] => [1,1,1] => 1110 => 0110 => 2

[[1,1,3,3]] => [2,2] => 1100 => 0100 => 1

[[2,2,3,3]] => [2,2] => 1100 => 0100 => 1

[[1,1,3],[3]] => [2,2] => 1100 => 0100 => 1

[[2,2,3],[3]] => [2,2] => 1100 => 0100 => 1

[[1,1],[3,3]] => [2,2] => 1100 => 0100 => 1

[[2,2],[3,3]] => [2,2] => 1100 => 0100 => 1

[[1,6]] => [1,1] => 110 => 010 => 1

[[2,6]] => [1,1] => 110 => 010 => 1

[[3,6]] => [1,1] => 110 => 010 => 1

[[4,6]] => [1,1] => 110 => 010 => 1

[[5,6]] => [1,1] => 110 => 010 => 1

[[1],[6]] => [1,1] => 110 => 010 => 1

[[2],[6]] => [1,1] => 110 => 010 => 1

[[3],[6]] => [1,1] => 110 => 010 => 1

[[4],[6]] => [1,1] => 110 => 010 => 1

[[5],[6]] => [1,1] => 110 => 010 => 1

[[1,1,5]] => [2,1] => 1010 => 0010 => 1

[[1,2,5]] => [1,1,1] => 1110 => 0110 => 2

[[1,3,5]] => [1,1,1] => 1110 => 0110 => 2

[[1,4,5]] => [1,1,1] => 1110 => 0110 => 2

[[1,5,5]] => [2,1] => 1010 => 0010 => 1

[[2,2,5]] => [2,1] => 1010 => 0010 => 1

[[2,3,5]] => [1,1,1] => 1110 => 0110 => 2

[[2,4,5]] => [1,1,1] => 1110 => 0110 => 2

[[2,5,5]] => [2,1] => 1010 => 0010 => 1

[[3,3,5]] => [2,1] => 1010 => 0010 => 1

[[3,4,5]] => [1,1,1] => 1110 => 0110 => 2

[[3,5,5]] => [2,1] => 1010 => 0010 => 1

[[4,4,5]] => [2,1] => 1010 => 0010 => 1

[[4,5,5]] => [2,1] => 1010 => 0010 => 1

[[1,1],[5]] => [2,1] => 1010 => 0010 => 1

[[1,2],[5]] => [1,1,1] => 1110 => 0110 => 2

[[1,5],[2]] => [1,1,1] => 1110 => 0110 => 2

[[1,3],[5]] => [1,1,1] => 1110 => 0110 => 2

[[1,5],[3]] => [1,1,1] => 1110 => 0110 => 2

[[1,4],[5]] => [1,1,1] => 1110 => 0110 => 2

[[1,5],[4]] => [1,1,1] => 1110 => 0110 => 2

[[1,5],[5]] => [2,1] => 1010 => 0010 => 1

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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

twist

Description

Return the binary word with first letter inverted.

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

weight

Description

The weight of a semistandard tableau as an integer partition.
The weight (or content) of a semistandard tableaux $T$ with maximal entry $m$ is the weak composition $(\alpha_1, \dots, \alpha_m)$ such that $\alpha_i$ is the number of letters $i$ occurring in $T$.
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.