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Identifier

St000945: 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)] => 16

[(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)] => 16

[(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)] => 16

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

[(1,5),(2,7),(3,4),(6,8)] => 16

[(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)] => 16

[(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)] => 16

[(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)] => 16

[(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)] => 16

[(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)] => 16

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

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Description

The number of matchings in the dihedral orbit of a perfect matching.
The dihedral orbit is induced by the dihedral symmetry of the underlying circular arrangement of points. In other words, this is the number of matchings that can be obtained by rotating or reflecting the given perfect matching.

Code

def statistic(m):
    n = m.size()
    l = set()
    m = tuple(sorted(tuple(sorted(p)) for p in m))
    l.add(m)
    l.add(tuple(sorted((n+1-j, n+1-i) for (i,j) in m)))
    for i in range(n):
        m = tuple(sorted(tuple(sorted(((i % n) + 1, (j % n) + 1))) for (i,j) in m))
        l.add(m)
        l.add(tuple(sorted((n+1-j, n+1-i) for (i,j) in m)))
    return len(l)

Created

Aug 22, 2017 at 14:41 by Christian Stump

Updated

Aug 22, 2017 at 21:17 by Martin Rubey