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Identifier

St001784: Set partitions ⟶ ℤ

Values

{{1}} => 1

{{1,2}} => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 2

{{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}} => 2

{{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 minimum of the smallest closer and the second element of the block containing 1 in a set partition.
A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as $1$ if $\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing $1$.

References

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

Code

def statistic(P):
    if not P:
        return
    r = None
    for b in P:
        b = sorted(b)
        if b[0] == 1:
            if len(b) == 1:
                return 1
            s = b[1]
        if len(b) > 1 and (r is None or r > b[-1]):
            r = b[-1]

    return min(r, s)

Created

Apr 07, 2022 at 09:28 by Martin Rubey

Updated

Apr 07, 2022 at 09:28 by Martin Rubey