- FindStat

Identifier

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[[1,1,2]] => [3] => 1000 => [1,4] => 2

[[1,2,2]] => [3] => 1000 => [1,4] => 2

[[2,2,2]] => [3] => 1000 => [1,4] => 2

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

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

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

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

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

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

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

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

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

[[1,1,3]] => [3] => 1000 => [1,4] => 2

[[1,2,3]] => [3] => 1000 => [1,4] => 2

[[1,3,3]] => [3] => 1000 => [1,4] => 2

[[2,2,3]] => [3] => 1000 => [1,4] => 2

[[2,3,3]] => [3] => 1000 => [1,4] => 2

[[3,3,3]] => [3] => 1000 => [1,4] => 2

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

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

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

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

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

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

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

[[1,1,1,2]] => [4] => 10000 => [1,5] => 2

[[1,1,2,2]] => [4] => 10000 => [1,5] => 2

[[1,2,2,2]] => [4] => 10000 => [1,5] => 2

[[2,2,2,2]] => [4] => 10000 => [1,5] => 2

[[1,1,1],[2]] => [3,1] => 10010 => [1,3,2] => 2

[[1,1,2],[2]] => [3,1] => 10010 => [1,3,2] => 2

[[1,2,2],[2]] => [3,1] => 10010 => [1,3,2] => 2

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

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

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

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

[[4,5]] => [2] => 100 => [1,3] => 2

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

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

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

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

[[4],[5]] => [1,1] => 110 => [1,1,2] => 3

[[1,1,4]] => [3] => 1000 => [1,4] => 2

[[1,2,4]] => [3] => 1000 => [1,4] => 2

[[1,3,4]] => [3] => 1000 => [1,4] => 2

[[1,4,4]] => [3] => 1000 => [1,4] => 2

[[2,2,4]] => [3] => 1000 => [1,4] => 2

[[2,3,4]] => [3] => 1000 => [1,4] => 2

[[2,4,4]] => [3] => 1000 => [1,4] => 2

[[3,3,4]] => [3] => 1000 => [1,4] => 2

[[3,4,4]] => [3] => 1000 => [1,4] => 2

[[4,4,4]] => [3] => 1000 => [1,4] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,1,1,3]] => [4] => 10000 => [1,5] => 2

[[1,1,2,3]] => [4] => 10000 => [1,5] => 2

[[1,1,3,3]] => [4] => 10000 => [1,5] => 2

[[1,2,2,3]] => [4] => 10000 => [1,5] => 2

[[1,2,3,3]] => [4] => 10000 => [1,5] => 2

[[1,3,3,3]] => [4] => 10000 => [1,5] => 2

[[2,2,2,3]] => [4] => 10000 => [1,5] => 2

[[2,2,3,3]] => [4] => 10000 => [1,5] => 2

[[2,3,3,3]] => [4] => 10000 => [1,5] => 2

[[3,3,3,3]] => [4] => 10000 => [1,5] => 2

[[1,1,1],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[1,1,2],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[1,1,3],[2]] => [3,1] => 10010 => [1,3,2] => 2

[[1,1,3],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[1,2,2],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[1,2,3],[2]] => [3,1] => 10010 => [1,3,2] => 2

[[1,2,3],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[1,3,3],[2]] => [3,1] => 10010 => [1,3,2] => 2

[[1,3,3],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[2,2,2],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[2,2,3],[3]] => [3,1] => 10010 => [1,3,2] => 2

[[2,3,3],[3]] => [3,1] => 10010 => [1,3,2] => 2

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

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

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

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

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Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Map

shape

Description

Return the shape of a tableau.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

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.