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Identifier

St001046: 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)] => 1

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

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

[(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)] => 1

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

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

[(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)] => 1

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

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

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

[(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)] => 1

[(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)] => 1

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

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

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

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

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

[(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)] => 1

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

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

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

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

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

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

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

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

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

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

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

[(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)] => 1

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

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

[(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)] => 2

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

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

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

[(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)] => 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)] => 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)] => 1

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

[(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)] => 1

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

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

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

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

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

[(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)] => 1

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

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Description

The maximal number of arcs nesting a given arc of a perfect matching.
This is also the largest weight of a down step in the histoire d'Hermite corresponding to the perfect matching.

Code

def statistic(m):
    w = 0
    active = []
    for i in range(1,m.size()+1):
        j = m.partner(i)
        if j < i:
            # i is a closer, j its opener
            w = max(w, len([1 for o in active if o < j]))
            active.remove(j)
        else:
            # i is an opener
            active.append(i)
    return w

Created

Nov 13, 2017 at 02:33 by Martin Rubey

Updated

May 14, 2018 at 21:07 by Martin Rubey