Identifier
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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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
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.