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Identifier

St000925: Set partitions ⟶ ℤ

Values

{{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}} => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is oeis:A099947.

Code

def statistic(M):
    """The number of topologically connected components of the arc
    diagram of a set partition."""

    C = dict()
    for b in M:
        for i in b:
            C[i] = frozenset(b)

    for (i1,j1), (i2,j2) in Subsets(M.arcs(), 2):
        if i1 < i2 < j1 < j2 or i2 < i1 < j2 < j1:
            if C[i1] != C[i2]:
                C[i1] = C[i1].union(C[i2])
                for a in C[i1]:
                    C[a] = C[i1]

    return len(set(c for c in C.values()))

Created

Aug 03, 2017 at 09:48 by Martin Rubey

Updated

Aug 03, 2017 at 15:05 by Martin Rubey