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Identifier

Mp00249: Set partitions —Callan switch⟶ Set partitions

Images

{{1}} => {{1}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Switch the first closer and the minimum of the smallest closer and the second element of the block containing 1 in a set partition.
More precisely, this involution implements a joint symmetry between St000971The smallest closer of a set partition. and St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition..

References

[1] Callan, D. An involution on set partitions arXiv:2204.02556

Properties

bijective, graded, involutive

Sage code

def mapping(P):
    """
    EXAMPLES::

        sage: mapping(SetPartition([[3],[4],[7],[8,5,2],[9,6,1]]))
        {{1, 3, 4, 9}, {2, 5, 8}, {6}, {7}}

        sage: mapping(SetPartition([[3],[4],[6,5,2],[7],[9,8,1]]))
        {{1, 3, 4, 8, 9}, {2, 5, 6}, {7}}


        sage: n = 10
        sage: all(mapping(mapping(P)) == P for P in SetPartitions(n))
        True

    """
    if not P:
        return P
    P = sorted((sorted(b) for b in P), key=max)

    r = None
    for b in P:
        if b[0] == 1:
            b1 = b
            if len(b1) == 1:
                y = 1
                break
            s = b[1]
        if len(b) > 1 and (r is None or r > b[-1]):
            r = b[-1]
    else:
        y = min(r, s)
    x = P[0][-1] # smallest max

    if x == y:
        result = P
    elif x < y:
        if r > s:
            i = 0
            while len(P[i]) == 1 and P[i][0] < s:
                i += 1
            initial_singletons = sum(P[:i], [])
            del b1[1]
            b1.extend(initial_singletons)
            result = P[i:] + [[s]]
        else:
            i = 0
            while len(P[i]) == 1:
                i += 1
            initial_singletons = sum(P[:i], [])
            b1.extend(initial_singletons)
            result = P[i:]
    else:
        if len(P[0]) == 1:
            initial_singletons = []
            while b1[1] < x:
                initial_singletons.append([b1[1]])
                del b1[1]
            b1.append(x)
            result = initial_singletons + P[1:]
        else:
            initial_singletons = []
            while b1[1] < x:
                initial_singletons.append([b1[1]])
                del b1[1]
            result = initial_singletons + P

    return SetPartition(result)

Weight

Created

Apr 12, 2022 at 15:54 by Martin Rubey

Updated

Oct 14, 2024 at 23:36 by Nadia Lafreniere