- FindStat

Identifier

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words

Images

[[1,2]] => [2] => [[1,2]] => 0

[[2,2]] => [2] => [[1,2]] => 0

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

[[1,3]] => [2] => [[1,2]] => 0

[[2,3]] => [2] => [[1,2]] => 0

[[3,3]] => [2] => [[1,2]] => 0

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

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

[[1,1,2]] => [3] => [[1,2,3]] => 00

[[1,2,2]] => [3] => [[1,2,3]] => 00

[[2,2,2]] => [3] => [[1,2,3]] => 00

[[1,1],[2]] => [2,1] => [[1,2],[3]] => 01

[[1,2],[2]] => [2,1] => [[1,2],[3]] => 01

[[1,4]] => [2] => [[1,2]] => 0

[[2,4]] => [2] => [[1,2]] => 0

[[3,4]] => [2] => [[1,2]] => 0

[[4,4]] => [2] => [[1,2]] => 0

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

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

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

[[1,1,3]] => [3] => [[1,2,3]] => 00

[[1,2,3]] => [3] => [[1,2,3]] => 00

[[1,3,3]] => [3] => [[1,2,3]] => 00

[[2,2,3]] => [3] => [[1,2,3]] => 00

[[2,3,3]] => [3] => [[1,2,3]] => 00

[[3,3,3]] => [3] => [[1,2,3]] => 00

[[1,1],[3]] => [2,1] => [[1,2],[3]] => 01

[[1,2],[3]] => [2,1] => [[1,2],[3]] => 01

[[1,3],[2]] => [2,1] => [[1,2],[3]] => 01

[[1,3],[3]] => [2,1] => [[1,2],[3]] => 01

[[2,2],[3]] => [2,1] => [[1,2],[3]] => 01

[[2,3],[3]] => [2,1] => [[1,2],[3]] => 01

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

[[1,1,1,2]] => [4] => [[1,2,3,4]] => 000

[[1,1,2,2]] => [4] => [[1,2,3,4]] => 000

[[1,2,2,2]] => [4] => [[1,2,3,4]] => 000

[[2,2,2,2]] => [4] => [[1,2,3,4]] => 000

[[1,1,1],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1,2],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,2,2],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1],[2,2]] => [2,2] => [[1,2],[3,4]] => 010

[[1,5]] => [2] => [[1,2]] => 0

[[2,5]] => [2] => [[1,2]] => 0

[[3,5]] => [2] => [[1,2]] => 0

[[4,5]] => [2] => [[1,2]] => 0

[[5,5]] => [2] => [[1,2]] => 0

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

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

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

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

[[1,1,4]] => [3] => [[1,2,3]] => 00

[[1,2,4]] => [3] => [[1,2,3]] => 00

[[1,3,4]] => [3] => [[1,2,3]] => 00

[[1,4,4]] => [3] => [[1,2,3]] => 00

[[2,2,4]] => [3] => [[1,2,3]] => 00

[[2,3,4]] => [3] => [[1,2,3]] => 00

[[2,4,4]] => [3] => [[1,2,3]] => 00

[[3,3,4]] => [3] => [[1,2,3]] => 00

[[3,4,4]] => [3] => [[1,2,3]] => 00

[[4,4,4]] => [3] => [[1,2,3]] => 00

[[1,1],[4]] => [2,1] => [[1,2],[3]] => 01

[[1,2],[4]] => [2,1] => [[1,2],[3]] => 01

[[1,4],[2]] => [2,1] => [[1,2],[3]] => 01

[[1,3],[4]] => [2,1] => [[1,2],[3]] => 01

[[1,4],[3]] => [2,1] => [[1,2],[3]] => 01

[[1,4],[4]] => [2,1] => [[1,2],[3]] => 01

[[2,2],[4]] => [2,1] => [[1,2],[3]] => 01

[[2,3],[4]] => [2,1] => [[1,2],[3]] => 01

[[2,4],[3]] => [2,1] => [[1,2],[3]] => 01

[[2,4],[4]] => [2,1] => [[1,2],[3]] => 01

[[3,3],[4]] => [2,1] => [[1,2],[3]] => 01

[[3,4],[4]] => [2,1] => [[1,2],[3]] => 01

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

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

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

[[1,1,1,3]] => [4] => [[1,2,3,4]] => 000

[[1,1,2,3]] => [4] => [[1,2,3,4]] => 000

[[1,1,3,3]] => [4] => [[1,2,3,4]] => 000

[[1,2,2,3]] => [4] => [[1,2,3,4]] => 000

[[1,2,3,3]] => [4] => [[1,2,3,4]] => 000

[[1,3,3,3]] => [4] => [[1,2,3,4]] => 000

[[2,2,2,3]] => [4] => [[1,2,3,4]] => 000

[[2,2,3,3]] => [4] => [[1,2,3,4]] => 000

[[2,3,3,3]] => [4] => [[1,2,3,4]] => 000

[[3,3,3,3]] => [4] => [[1,2,3,4]] => 000

[[1,1,1],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1,2],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1,3],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1,3],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,2,2],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,2,3],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,2,3],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,3,3],[2]] => [3,1] => [[1,2,3],[4]] => 001

[[1,3,3],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[2,2,2],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[2,2,3],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[2,3,3],[3]] => [3,1] => [[1,2,3],[4]] => 001

[[1,1],[2,3]] => [2,2] => [[1,2],[3,4]] => 010

[[1,1],[3,3]] => [2,2] => [[1,2],[3,4]] => 010

[[1,2],[2,3]] => [2,2] => [[1,2],[3,4]] => 010

[[1,2],[3,3]] => [2,2] => [[1,2],[3,4]] => 010

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Map

shape

Description

Return the shape of a tableau.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.

Map

descent word

Description

The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.