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Identifier

St000780: 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)] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The size of the orbit under rotation of a perfect matching.
The number of orbits is given in [1].

References

[1] Number of chord diagrams with n chords; number of pairings on a necklace. OEIS:A007769

Code

def statistic(m):
    n = m.size()
    def rot(x):
        return PerfectMatching([((i % n) + 1, (j % n) + 1) for i, j in x])
    DS = DiscreteDynamicalSystem(m.parent(), rot)
    return len(DS.orbit(m))

def statistic_alternative(m):
    m = sorted([sorted((i,j)) for (i,j) in m])
    m2 = m
    s = 1
    n = 2*len(m)
    for i in range(n):
        m2 = sorted([sorted(((i % n) + 1, (j % n) + 1)) for (i,j) in m2])
        if m2 == m:
            return s
        else:
            s += 1

Created

Apr 17, 2017 at 19:13 by Martin Rubey

Updated

Feb 02, 2023 at 10:44 by Martin Rubey