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Identifier

St000754: Perfect matchings ⟶ ℤ

Values

[(1,2)] => 0

[(1,2),(3,4)] => 0

[(1,3),(2,4)] => 0

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

[(1,2),(3,4),(5,6)] => 0

[(1,3),(2,4),(5,6)] => 0

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

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

[(1,6),(2,3),(4,5)] => 0

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

[(1,5),(2,4),(3,6)] => 0

[(1,4),(2,5),(3,6)] => 0

[(1,3),(2,5),(4,6)] => 0

[(1,2),(3,5),(4,6)] => 0

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

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

[(1,4),(2,6),(3,5)] => 0

[(1,5),(2,6),(3,4)] => 0

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

[(1,2),(3,4),(5,6),(7,8)] => 0

[(1,3),(2,4),(5,6),(7,8)] => 0

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

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

[(1,6),(2,3),(4,5),(7,8)] => 0

[(1,7),(2,3),(4,5),(6,8)] => 0

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

[(1,8),(2,4),(3,5),(6,7)] => 0

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

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

[(1,5),(2,4),(3,6),(7,8)] => 0

[(1,4),(2,5),(3,6),(7,8)] => 0

[(1,3),(2,5),(4,6),(7,8)] => 0

[(1,2),(3,5),(4,6),(7,8)] => 0

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

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

[(1,4),(2,6),(3,5),(7,8)] => 0

[(1,5),(2,6),(3,4),(7,8)] => 0

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

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

[(1,8),(2,5),(3,4),(6,7)] => 0

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

[(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)] => 0

[(1,4),(2,7),(3,5),(6,8)] => 0

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

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

[(1,2),(3,8),(4,5),(6,7)] => 0

[(1,3),(2,8),(4,5),(6,7)] => 0

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

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

[(1,6),(2,8),(3,4),(5,7)] => 0

[(1,7),(2,8),(3,4),(5,6)] => 0

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

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

[(1,7),(2,8),(3,5),(4,6)] => 0

[(1,6),(2,8),(3,5),(4,7)] => 0

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

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

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

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

[(1,2),(3,7),(4,6),(5,8)] => 0

[(1,3),(2,7),(4,6),(5,8)] => 0

[(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)] => 2

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

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

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

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

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

[(1,5),(2,6),(3,7),(4,8)] => 0

[(1,4),(2,6),(3,7),(5,8)] => 0

[(1,3),(2,6),(4,7),(5,8)] => 0

[(1,2),(3,6),(4,7),(5,8)] => 0

[(1,2),(3,5),(4,7),(6,8)] => 0

[(1,3),(2,5),(4,7),(6,8)] => 0

[(1,4),(2,5),(3,7),(6,8)] => 0

[(1,5),(2,4),(3,7),(6,8)] => 0

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

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

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

[(1,8),(2,3),(4,6),(5,7)] => 0

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

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

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

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

[(1,3),(2,4),(5,7),(6,8)] => 0

[(1,2),(3,4),(5,7),(6,8)] => 0

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

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

[(1,4),(2,3),(5,8),(6,7)] => 0

[(1,5),(2,3),(4,8),(6,7)] => 0

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

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

[(1,8),(2,3),(4,7),(5,6)] => 0

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

[(1,7),(2,4),(3,8),(5,6)] => 0

[(1,6),(2,4),(3,8),(5,7)] => 0

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

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

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Description

The Grundy value for the game of removing nestings in a perfect matching.
A move consists of choosing a nesting, that is two pairs $(a,d)$ and $(b,c)$ with $a < b < c < d$ and replacing them with the two pairs $(a,b)$ and $(c,d)$. The player facing a non-nesting matching looses.

Code

@cached_function
def statistic(w):
    def children(m):
        for (a,b),(c,d) in m.nestings():
            m_new = list(m)
            m_new.remove((a,b))
            m_new.remove((c,d))
            A = min(a,b)
            B = max(a,b)
            C = min(c,d)
            D = max(c,d)
            if C < A:
                (A,B),(C,D) = (C,D),(A,B)
            yield PerfectMatching(m_new + [(A,C), (B,D)])

    l = [statistic(v) for v in children(w)]
    i = 0
    while i in l:
        i += 1
    return i

Created

Apr 07, 2017 at 21:48 by Martin Rubey

Updated

Apr 08, 2017 at 08:58 by Martin Rubey