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Identifier

St001832: Perfect matchings ⟶ ℤ

Values

[(1,2)] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[(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)] => 10

[(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)] => 2

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

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

[(1,4),(2,7),(3,6),(5,8)] => 12

[(1,5),(2,7),(3,6),(4,8)] => 13

[(1,6),(2,7),(3,5),(4,8)] => 10

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

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

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

[(1,7),(2,5),(3,6),(4,8)] => 10

[(1,6),(2,5),(3,7),(4,8)] => 13

[(1,5),(2,6),(3,7),(4,8)] => 14

[(1,4),(2,6),(3,7),(5,8)] => 13

[(1,3),(2,6),(4,7),(5,8)] => 10

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

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

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

[(1,4),(2,5),(3,7),(6,8)] => 10

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

[(1,6),(2,4),(3,7),(5,8)] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of non-crossing perfect matchings in the chord expansion of a perfect matching.
Given a perfect matching, we obtain a formal sum of non-crossing perfect matchings by replacing recursively every matching $M$ that has a crossing $(a, c), (b, d)$ with $a < b < c < d$ with the sum of the two matchings $(M\setminus \{(a,c), (b,d)\})\cup \{(a,b), (c,d)\}$ and $(M\setminus \{(a,c), (b,d)\})\cup \{(a,d), (b,c)\}$.
This statistic is the number of distinct non-crossing perfect matchings in the formal sum.

Code

def expand(M, ac, bd):
    a, c = ac
    b, d = bd
    m = list(M)
    m.remove(frozenset(ac))
    m.remove(frozenset(bd))
    m1 = m + [frozenset([a, b]), frozenset([c, d])]
    m2 = m + [frozenset([a, d]), frozenset([b, c])]
    return PerfectMatching(m1), PerfectMatching(m2)

def expansion(M):
    input = {M: 1}
    output = {}
    while input:
        m, e = input.popitem()
        try:
            ac, bd = next(m.crossings_iterator())
        except StopIteration:
            output[m] = output.get(m, 0) + e
        else:
            m1, m2 = expand(m, ac, bd)
            input[m1] = input.get(m1, 0) + e
            input[m2] = input.get(m2, 0) + e
    return output

def statistic(M):
    return len(expansion(M))

Created

Sep 01, 2022 at 00:00 by Martin Rubey

Updated

Sep 01, 2022 at 00:00 by Martin Rubey