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Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001781: Set partitions ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + q$

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

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

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

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

Description

The interlacing number of a set partition.
Let

$\pi$ be a set partition of

$\{1,\dots,n\}$ with

$k$ blocks. To each block of

$\pi$ we add the element

$\infty$ , which is larger than

$n$ . Then, an interlacing of

$\pi$ is a pair of blocks

$B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and

$C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index

$1\leq i\leq \min(b, c)$ , such that

$B_i < C_i < B_{i+1} < C_{i+1}$ .

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

rows

Description

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