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Identifier

St000793: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 1

{{1},{2}} => 2

{{1,2,3}} => 1

{{1,2},{3}} => 2

{{1,3},{2}} => 2

{{1},{2,3}} => 2

{{1},{2},{3}} => 2

{{1,2,3,4}} => 1

{{1,2,3},{4}} => 2

{{1,2,4},{3}} => 2

{{1,2},{3,4}} => 2

{{1,2},{3},{4}} => 2

{{1,3,4},{2}} => 2

{{1,3},{2,4}} => 2

{{1,3},{2},{4}} => 2

{{1,4},{2,3}} => 2

{{1},{2,3,4}} => 2

{{1},{2,3},{4}} => 2

{{1,4},{2},{3}} => 3

{{1},{2,4},{3}} => 2

{{1},{2},{3,4}} => 2

{{1},{2},{3},{4}} => 2

{{1,2,3,4,5}} => 1

{{1,2,3,4},{5}} => 2

{{1,2,3,5},{4}} => 2

{{1,2,3},{4,5}} => 2

{{1,2,3},{4},{5}} => 2

{{1,2,4,5},{3}} => 2

{{1,2,4},{3,5}} => 2

{{1,2,4},{3},{5}} => 2

{{1,2,5},{3,4}} => 2

{{1,2},{3,4,5}} => 2

{{1,2},{3,4},{5}} => 2

{{1,2,5},{3},{4}} => 3

{{1,2},{3,5},{4}} => 2

{{1,2},{3},{4,5}} => 2

{{1,2},{3},{4},{5}} => 2

{{1,3,4,5},{2}} => 2

{{1,3,4},{2,5}} => 2

{{1,3,4},{2},{5}} => 2

{{1,3,5},{2,4}} => 2

{{1,3},{2,4,5}} => 2

{{1,3},{2,4},{5}} => 2

{{1,3,5},{2},{4}} => 2

{{1,3},{2,5},{4}} => 3

{{1,3},{2},{4,5}} => 2

{{1,3},{2},{4},{5}} => 2

{{1,4,5},{2,3}} => 2

{{1,4},{2,3,5}} => 2

{{1,4},{2,3},{5}} => 2

{{1,5},{2,3,4}} => 2

{{1},{2,3,4,5}} => 2

{{1},{2,3,4},{5}} => 2

{{1,5},{2,3},{4}} => 3

{{1},{2,3,5},{4}} => 2

{{1},{2,3},{4,5}} => 2

{{1},{2,3},{4},{5}} => 2

{{1,4,5},{2},{3}} => 3

{{1,4},{2,5},{3}} => 3

{{1,4},{2},{3,5}} => 3

{{1,4},{2},{3},{5}} => 3

{{1,5},{2,4},{3}} => 2

{{1},{2,4,5},{3}} => 2

{{1},{2,4},{3,5}} => 2

{{1},{2,4},{3},{5}} => 2

{{1,5},{2},{3,4}} => 3

{{1},{2,5},{3,4}} => 2

{{1},{2},{3,4,5}} => 2

{{1},{2},{3,4},{5}} => 2

{{1,5},{2},{3},{4}} => 3

{{1},{2,5},{3},{4}} => 3

{{1},{2},{3,5},{4}} => 2

{{1},{2},{3},{4,5}} => 2

{{1},{2},{3},{4},{5}} => 2

{{1,2,3,4,5,6}} => 1

{{1,2,3,4,5},{6}} => 2

{{1,2,3,4,6},{5}} => 2

{{1,2,3,4},{5,6}} => 2

{{1,2,3,4},{5},{6}} => 2

{{1,2,3,5,6},{4}} => 2

{{1,2,3,5},{4,6}} => 2

{{1,2,3,5},{4},{6}} => 2

{{1,2,3,6},{4,5}} => 2

{{1,2,3},{4,5,6}} => 2

{{1,2,3},{4,5},{6}} => 2

{{1,2,3,6},{4},{5}} => 3

{{1,2,3},{4,6},{5}} => 2

{{1,2,3},{4},{5,6}} => 2

{{1,2,3},{4},{5},{6}} => 2

{{1,2,4,5,6},{3}} => 2

{{1,2,4,5},{3,6}} => 2

{{1,2,4,5},{3},{6}} => 2

{{1,2,4,6},{3,5}} => 2

{{1,2,4},{3,5,6}} => 2

{{1,2,4},{3,5},{6}} => 2

{{1,2,4,6},{3},{5}} => 2

{{1,2,4},{3,6},{5}} => 3

{{1,2,4},{3},{5,6}} => 2

{{1,2,4},{3},{5},{6}} => 2

{{1,2,5,6},{3,4}} => 2

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Description

The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.

References

[1] Benkart, G., Halverson, T., Harman, N. Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups arXiv:1605.06543

Code

def from_partition(p, n):
    """
    sage: G = from_partition([[1,3,4],[2]], 4); G
      0  0  1
      1  0
      0
    sage: G.in_labels()
    [[3], [2], [2], [2], [2], [3], [3]]
    sage: G.out_labels()
    [[3], [3, 1], [3], [3, 1], [3], [4], [3]]

    sage: G = from_partition([[3,4],[1,2]], 4); G
    0  0  1
    0  0
    1
    sage: G.in_labels()
    [[3], [3], [2], [2], [2], [3], [3]]
    sage: G.out_labels()
    [[3], [4], [3], [3, 1], [3], [4], [3]]

    sage: G = from_partition([[3,4],[2],[1]], 4); G
    0  0  1
    0  0
    0
    sage: G.in_labels()
    [[3], [2], [1], [1], [2], [3], [3]]
    sage: G.out_labels()
    [[3], [3, 1], [2, 1], [3, 1], [3], [4], [3]]

    """
    p = SetPartition(p)
    r = p.size()
    filling = {(i-1, r-j): 1 for (i,j) in p.arcs()}
    shape = Partition([r-i-1 for i in range(r-1)])
    labels = [[n-1]]
    la = n-1
    O = [i for i,_ in p.arcs()]
    C = [j for _,j in p.arcs()]
    for j in range(2,r+1):
        if j not in C:
            la -= 1
        labels += [[la]]
    for i in range(1,r):
        if i not in O:
            la += 1
        labels += [[la]]
    return GrowthDiagram.rules.RSK()(filling, shape, labels)


def statistic(p):
    return max(len(mu) for mu in from_partition(p, p.size()).out_labels())

Created

May 10, 2017 at 22:55 by Martin Rubey

Updated

Nov 11, 2021 at 12:50 by Martin Rubey