- FindStat

Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000585: Set partitions ⟶ ℤ

Values

[2] => [[1,2]] => {{1,2}} => 0
[1,1] => [[1],[2]] => {{1},{2}} => 0
[3] => [[1,2,3]] => {{1,2,3}} => 0
[2,1] => [[1,3],[2]] => {{1,3},{2}} => 1
[1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 0
[4] => [[1,2,3,4]] => {{1,2,3,4}} => 0
[3,1] => [[1,3,4],[2]] => {{1,3,4},{2}} => 1
[2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 0
[2,1,1] => [[1,4],[2],[3]] => {{1,4},{2},{3}} => 2
[1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 0
[5] => [[1,2,3,4,5]] => {{1,2,3,4,5}} => 0
[4,1] => [[1,3,4,5],[2]] => {{1,3,4,5},{2}} => 1
[3,2] => [[1,2,5],[3,4]] => {{1,2,5},{3,4}} => 1
[3,1,1] => [[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => 2
[2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => 2
[2,1,1,1] => [[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => 3
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 0
[6] => [[1,2,3,4,5,6]] => {{1,2,3,4,5,6}} => 0
[5,1] => [[1,3,4,5,6],[2]] => {{1,3,4,5,6},{2}} => 1
[4,2] => [[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => 1
[4,1,1] => [[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => 2
[3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 0
[3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => 3
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 0
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => {{1,4},{2,6},{3},{5}} => 4
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => {{1,6},{2},{3},{4},{5}} => 4
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => 0
[7] => [[1,2,3,4,5,6,7]] => {{1,2,3,4,5,6,7}} => 0
[6,1] => [[1,3,4,5,6,7],[2]] => {{1,3,4,5,6,7},{2}} => 1
[5,2] => [[1,2,5,6,7],[3,4]] => {{1,2,5,6,7},{3,4}} => 1
[5,1,1] => [[1,4,5,6,7],[2],[3]] => {{1,4,5,6,7},{2},{3}} => 2
[4,3] => [[1,2,3,7],[4,5,6]] => {{1,2,3,7},{4,5,6}} => 1
[4,2,1] => [[1,3,6,7],[2,5],[4]] => {{1,3,6,7},{2,5},{4}} => 3
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => {{1,5,6,7},{2},{3},{4}} => 3
[3,3,1] => [[1,3,4],[2,6,7],[5]] => {{1,3,4},{2,6,7},{5}} => 2
[3,2,2] => [[1,2,7],[3,4],[5,6]] => {{1,2,7},{3,4},{5,6}} => 2
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => {{1,4,7},{2,6},{3},{5}} => 5
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => {{1,6,7},{2},{3},{4},{5}} => 4
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => {{1,3},{2,5},{4,7},{6}} => 3
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => {{1,5},{2,7},{3},{4},{6}} => 6
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => {{1,7},{2},{3},{4},{5},{6}} => 5
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => {{1},{2},{3},{4},{5},{6},{7}} => 0
[8] => [[1,2,3,4,5,6,7,8]] => {{1,2,3,4,5,6,7,8}} => 0
[7,1] => [[1,3,4,5,6,7,8],[2]] => {{1,3,4,5,6,7,8},{2}} => 1
[6,2] => [[1,2,5,6,7,8],[3,4]] => {{1,2,5,6,7,8},{3,4}} => 1
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => {{1,4,5,6,7,8},{2},{3}} => 2

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$F_{2} = 2$

$F_{3} = 2 + q$

$F_{4} = 3 + q + q^{2}$

$F_{5} = 2 + 2\ q + 2\ q^{2} + q^{3}$

$F_{6} = 4 + 2\ q + q^{2} + 2\ q^{3} + 2\ q^{4}$

$F_{7} = 2 + 3\ q + 3\ q^{2} + 3\ q^{3} + q^{4} + 2\ q^{5} + q^{6}$

Description

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.

Map

rows

Description

The set partition whose blocks are the rows of the tableau.

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.