- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000502: Set partitions ⟶ ℤ

Values

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

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$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2}$

$F_{4} = 1 + 2\ q + q^{2} + q^{3}$

$F_{5} = 1 + 2\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{6} = 1 + 3\ q + 2\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{7} = 1 + 3\ q + 3\ q^{2} + 3\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

Description

The number of successions of a set partitions.
This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

Map

rows

Description

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

Map

Yip

Description

A transformation of set partitions due to Yip.
Return the set partition of

$\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection

$\psi$ .