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Identifier

St000924: Perfect matchings ⟶ ℤ

Values

[(1,2)] => 1

[(1,2),(3,4)] => 2

[(1,3),(2,4)] => 1

[(1,4),(2,3)] => 2

[(1,2),(3,4),(5,6)] => 3

[(1,3),(2,4),(5,6)] => 2

[(1,4),(2,3),(5,6)] => 3

[(1,5),(2,3),(4,6)] => 2

[(1,6),(2,3),(4,5)] => 3

[(1,6),(2,4),(3,5)] => 2

[(1,5),(2,4),(3,6)] => 1

[(1,4),(2,5),(3,6)] => 1

[(1,3),(2,5),(4,6)] => 1

[(1,2),(3,5),(4,6)] => 2

[(1,2),(3,6),(4,5)] => 3

[(1,3),(2,6),(4,5)] => 2

[(1,4),(2,6),(3,5)] => 1

[(1,5),(2,6),(3,4)] => 2

[(1,6),(2,5),(3,4)] => 3

[(1,2),(3,4),(5,6),(7,8)] => 4

[(1,3),(2,4),(5,6),(7,8)] => 3

[(1,4),(2,3),(5,6),(7,8)] => 4

[(1,5),(2,3),(4,6),(7,8)] => 3

[(1,6),(2,3),(4,5),(7,8)] => 4

[(1,7),(2,3),(4,5),(6,8)] => 3

[(1,8),(2,3),(4,5),(6,7)] => 4

[(1,8),(2,4),(3,5),(6,7)] => 3

[(1,7),(2,4),(3,5),(6,8)] => 2

[(1,6),(2,4),(3,5),(7,8)] => 3

[(1,5),(2,4),(3,6),(7,8)] => 2

[(1,4),(2,5),(3,6),(7,8)] => 2

[(1,3),(2,5),(4,6),(7,8)] => 2

[(1,2),(3,5),(4,6),(7,8)] => 3

[(1,2),(3,6),(4,5),(7,8)] => 4

[(1,3),(2,6),(4,5),(7,8)] => 3

[(1,4),(2,6),(3,5),(7,8)] => 2

[(1,5),(2,6),(3,4),(7,8)] => 3

[(1,6),(2,5),(3,4),(7,8)] => 4

[(1,7),(2,5),(3,4),(6,8)] => 3

[(1,8),(2,5),(3,4),(6,7)] => 4

[(1,8),(2,6),(3,4),(5,7)] => 3

[(1,7),(2,6),(3,4),(5,8)] => 2

[(1,6),(2,7),(3,4),(5,8)] => 2

[(1,5),(2,7),(3,4),(6,8)] => 2

[(1,4),(2,7),(3,5),(6,8)] => 1

[(1,3),(2,7),(4,5),(6,8)] => 2

[(1,2),(3,7),(4,5),(6,8)] => 3

[(1,2),(3,8),(4,5),(6,7)] => 4

[(1,3),(2,8),(4,5),(6,7)] => 3

[(1,4),(2,8),(3,5),(6,7)] => 2

[(1,5),(2,8),(3,4),(6,7)] => 3

[(1,6),(2,8),(3,4),(5,7)] => 2

[(1,7),(2,8),(3,4),(5,6)] => 3

[(1,8),(2,7),(3,4),(5,6)] => 4

[(1,8),(2,7),(3,5),(4,6)] => 3

[(1,7),(2,8),(3,5),(4,6)] => 2

[(1,6),(2,8),(3,5),(4,7)] => 1

[(1,5),(2,8),(3,6),(4,7)] => 1

[(1,4),(2,8),(3,6),(5,7)] => 1

[(1,3),(2,8),(4,6),(5,7)] => 2

[(1,2),(3,8),(4,6),(5,7)] => 3

[(1,2),(3,7),(4,6),(5,8)] => 2

[(1,3),(2,7),(4,6),(5,8)] => 1

[(1,4),(2,7),(3,6),(5,8)] => 1

[(1,5),(2,7),(3,6),(4,8)] => 1

[(1,6),(2,7),(3,5),(4,8)] => 1

[(1,7),(2,6),(3,5),(4,8)] => 1

[(1,8),(2,6),(3,5),(4,7)] => 2

[(1,8),(2,5),(3,6),(4,7)] => 2

[(1,7),(2,5),(3,6),(4,8)] => 1

[(1,6),(2,5),(3,7),(4,8)] => 1

[(1,5),(2,6),(3,7),(4,8)] => 1

[(1,4),(2,6),(3,7),(5,8)] => 1

[(1,3),(2,6),(4,7),(5,8)] => 1

[(1,2),(3,6),(4,7),(5,8)] => 2

[(1,2),(3,5),(4,7),(6,8)] => 2

[(1,3),(2,5),(4,7),(6,8)] => 1

[(1,4),(2,5),(3,7),(6,8)] => 1

[(1,5),(2,4),(3,7),(6,8)] => 1

[(1,6),(2,4),(3,7),(5,8)] => 1

[(1,7),(2,4),(3,6),(5,8)] => 1

[(1,8),(2,4),(3,6),(5,7)] => 2

[(1,8),(2,3),(4,6),(5,7)] => 3

[(1,7),(2,3),(4,6),(5,8)] => 2

[(1,6),(2,3),(4,7),(5,8)] => 2

[(1,5),(2,3),(4,7),(6,8)] => 2

[(1,4),(2,3),(5,7),(6,8)] => 3

[(1,3),(2,4),(5,7),(6,8)] => 2

[(1,2),(3,4),(5,7),(6,8)] => 3

[(1,2),(3,4),(5,8),(6,7)] => 4

[(1,3),(2,4),(5,8),(6,7)] => 3

[(1,4),(2,3),(5,8),(6,7)] => 4

[(1,5),(2,3),(4,8),(6,7)] => 3

[(1,6),(2,3),(4,8),(5,7)] => 2

[(1,7),(2,3),(4,8),(5,6)] => 3

[(1,8),(2,3),(4,7),(5,6)] => 4

[(1,8),(2,4),(3,7),(5,6)] => 3

[(1,7),(2,4),(3,8),(5,6)] => 2

[(1,6),(2,4),(3,8),(5,7)] => 1

[(1,5),(2,4),(3,8),(6,7)] => 2

[(1,4),(2,5),(3,8),(6,7)] => 2

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Description

The number of topologically connected components of a perfect matching.
For example, the perfect matching $\{\{1,4\},\{2,3\}\}$ has the two connected components $\{1,4\}$ and $\{2,3\}$.
The number of perfect matchings with only one block is oeis:A000699.

References

[1] Gil, J. B., McNamara, P. R. W., Tirrell, J. O., Weiner, M. D. From Dyck paths to standard Young tableaux arXiv:1708.00513

Code

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

    C = dict()
    for a, b in M:
        if a > b:
            a,b = b,a
        C[a] = frozenset([a,b])
        C[b] = C[a]

    for (i,j), (k,l) in M.crossings():
        if C[i] != C[k]:
            C[i] = C[i].union(C[k])
            for a in C[i]:
                C[a] = C[i]

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

Created

Aug 03, 2017 at 09:34 by Martin Rubey

Updated

Aug 03, 2017 at 15:08 by Martin Rubey