- FindStat

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Identifier

St000729: Set partitions ⟶ ℤ

Values

{{1,2}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The minimal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).

Code

def statistic(SP):
    SP = sorted( sorted(B) for B in SP )
    arcs = sorted( sum( [[(B[i],B[i+1]) for i in range(len(B)-1)] for B in SP],[] ) )
    if arcs == []:
        return sum(len(B) for B in SP)
    else:
        return min( j-i for i,j in arcs )

Created

Mar 28, 2017 at 16:35 by Christian Stump

Updated

Mar 28, 2017 at 18:04 by Christian Stump