- FindStat

Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤ

Values

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

{{1},{2}} => [1,1] => [[1],[2]] => 1 => 1

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

{{1,2},{3}} => [2,1] => [[1,3],[2]] => 10 => 1

{{1,3},{2}} => [2,1] => [[1,3],[2]] => 10 => 1

{{1},{2,3}} => [2,1] => [[1,3],[2]] => 10 => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4}} => [1,1,1,1] => [[1],[2],[3],[4]] => 111 => 1

{{1,2,3,4,5}} => [5] => [[1,2,3,4,5]] => 0000 => 5

{{1,2,3,4},{5}} => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

{{1,2,3,5},{4}} => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

{{1,2,3},{4,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,2,3},{4},{5}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,2,4,5},{3}} => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

{{1,2,4},{3,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,2,4},{3},{5}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,2,5},{3,4}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,2},{3,4,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,2},{3,4},{5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,2,5},{3},{4}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,2},{3,5},{4}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,2},{3},{4,5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,2},{3},{4},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,3,4,5},{2}} => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

{{1,3,4},{2,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,3,4},{2},{5}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,3,5},{2,4}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,3},{2,4,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,3},{2,4},{5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,3,5},{2},{4}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,3},{2,5},{4}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,3},{2},{4,5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,3},{2},{4},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,4,5},{2,3}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,4},{2,3,5}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1,4},{2,3},{5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,5},{2,3,4}} => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

{{1},{2,3,4,5}} => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

{{1},{2,3,4},{5}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,5},{2,3},{4}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2,3,5},{4}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1},{2,3},{4,5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2,3},{4},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,4,5},{2},{3}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1,4},{2,5},{3}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,4},{2},{3,5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1,4},{2},{3},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,5},{2,4},{3}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2,4,5},{3}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1},{2,4},{3,5}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2,4},{3},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,5},{2},{3,4}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2,5},{3,4}} => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 1

{{1},{2},{3,4,5}} => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 1

{{1},{2},{3,4},{5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1,5},{2},{3},{4}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1},{2,5},{3},{4}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1},{2},{3,5},{4}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1},{2},{3},{4,5}} => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 1

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1111 => 1

{{1,2,3,4,5,6}} => [6] => [[1,2,3,4,5,6]] => 00000 => 6

{{1,2,3,4,5},{6}} => [5,1] => [[1,3,4,5,6],[2]] => 10000 => 1

{{1,2,3,4,6},{5}} => [5,1] => [[1,3,4,5,6],[2]] => 10000 => 1

{{1,2,3,4},{5,6}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,3,4},{5},{6}} => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 1

{{1,2,3,5,6},{4}} => [5,1] => [[1,3,4,5,6],[2]] => 10000 => 1

{{1,2,3,5},{4,6}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,3,5},{4},{6}} => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 1

{{1,2,3,6},{4,5}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,3},{4,5,6}} => [3,3] => [[1,2,3],[4,5,6]] => 00100 => 3

{{1,2,3},{4,5},{6}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,3,6},{4},{5}} => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 1

{{1,2,3},{4,6},{5}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,3},{4},{5,6}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 11100 => 1

{{1,2,4,5,6},{3}} => [5,1] => [[1,3,4,5,6],[2]] => 10000 => 1

{{1,2,4,5},{3,6}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,4,5},{3},{6}} => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 1

{{1,2,4,6},{3,5}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,4},{3,5,6}} => [3,3] => [[1,2,3],[4,5,6]] => 00100 => 3

{{1,2,4},{3,5},{6}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,4,6},{3},{5}} => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 1

{{1,2,4},{3,6},{5}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,4},{3},{5,6}} => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 1

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 11100 => 1

{{1,2,5,6},{3,4}} => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

{{1,2,5},{3,4,6}} => [3,3] => [[1,2,3],[4,5,6]] => 00100 => 3

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Description

The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.

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 $1\leq i < n$.

Map

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.