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Identifier

St000583: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 0

{{1},{2}} => 0

{{1,2,3}} => 0

{{1,2},{3}} => 0

{{1,3},{2}} => 0

{{1},{2,3}} => 0

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

{{1,2,3,4}} => 0

{{1,2,3},{4}} => 0

{{1,2,4},{3}} => 0

{{1,2},{3,4}} => 0

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

{{1,3,4},{2}} => 0

{{1,3},{2,4}} => 0

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

{{1,4},{2,3}} => 0

{{1},{2,3,4}} => 0

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

{{1,4},{2},{3}} => 0

{{1},{2,4},{3}} => 0

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

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

{{1,2,3,4,5}} => 0

{{1,2,3,4},{5}} => 0

{{1,2,3,5},{4}} => 0

{{1,2,3},{4,5}} => 0

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

{{1,2,4,5},{3}} => 0

{{1,2,4},{3,5}} => 0

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

{{1,2,5},{3,4}} => 0

{{1,2},{3,4,5}} => 0

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

{{1,2,5},{3},{4}} => 0

{{1,2},{3,5},{4}} => 0

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

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

{{1,3,4,5},{2}} => 0

{{1,3,4},{2,5}} => 0

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

{{1,3,5},{2,4}} => 0

{{1,3},{2,4,5}} => 0

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

{{1,3,5},{2},{4}} => 0

{{1,3},{2,5},{4}} => 0

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

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

{{1,4,5},{2,3}} => 0

{{1,4},{2,3,5}} => 0

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

{{1,5},{2,3,4}} => 0

{{1},{2,3,4,5}} => 0

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

{{1,5},{2,3},{4}} => 0

{{1},{2,3,5},{4}} => 0

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

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

{{1,4,5},{2},{3}} => 0

{{1,4},{2,5},{3}} => 0

{{1,4},{2},{3,5}} => 0

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

{{1,5},{2,4},{3}} => 0

{{1},{2,4,5},{3}} => 0

{{1},{2,4},{3,5}} => 0

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

{{1,5},{2},{3,4}} => 0

{{1},{2,5},{3,4}} => 0

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

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

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

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

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

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

{{1},{2},{3},{4},{5}} => 10

{{1,2,3,4,5,6}} => 0

{{1,2,3,4,5},{6}} => 0

{{1,2,3,4,6},{5}} => 0

{{1,2,3,4},{5,6}} => 0

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

{{1,2,3,5,6},{4}} => 0

{{1,2,3,5},{4,6}} => 0

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

{{1,2,3,6},{4,5}} => 0

{{1,2,3},{4,5,6}} => 0

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

{{1,2,3,6},{4},{5}} => 0

{{1,2,3},{4,6},{5}} => 0

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

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

{{1,2,4,5,6},{3}} => 0

{{1,2,4,5},{3,6}} => 0

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

{{1,2,4,6},{3,5}} => 0

{{1,2,4},{3,5,6}} => 0

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

{{1,2,4,6},{3},{5}} => 0

{{1,2,4},{3,6},{5}} => 0

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

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

{{1,2,5,6},{3,4}} => 0

>>> Load all 1503 entries. <<<

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Description

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal.

References

[1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Closed expressions for averages of set partition statistics MathSciNet:3338726 arXiv:1304.4309

Code

def statistic(pi):
    return len(pattern_occurrences(pi, [[1], [2], [3]], [3], [1, 2], [], []))

def pattern_occurrences(pi, P, First, Last, Arcs, Consecutives):
    """We assume that pi is a SetPartition of {1,2,...,n} and P is a
    SetPartition of {1,2,...,k}.
    """
    occurrences = []
    pi = SetPartition(pi)
    P = SetPartition(P)

    openers = [min(B) for B in pi]
    closers = [max(B) for B in pi]
    pi_sorted = sorted([sorted(b) for b in pi])
    edges = [(a,b) for B in pi_sorted for (a,b) in zip(B, B[1:])]

    for s in Subsets(pi.base_set(), P.size()):
        s = sorted(s)
        pi_r = pi.restriction(s)
        if pi_r.standardization() != P:
            continue

        X = pi_r.base_set()

        if any(X[i-1] not in openers for i in First):
            continue

        if any(X[i-1] not in closers for i in Last):
            continue

        if any((X[i-1], X[j-1]) not in edges for (i,j) in Arcs):
            continue

        if any(abs(X[i-1]-X[j-1]) != 1 for (i,j) in Consecutives):
            continue

        occurrences += [s]
    return occurrences

Created

Aug 11, 2016 at 14:50 by Martin Rubey

Updated

Apr 01, 2018 at 21:29 by Martin Rubey