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Identifier

St001589: Perfect matchings ⟶ ℤ

Values

[(1,2)] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[(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)] => 2

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

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

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

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

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

[(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)] => 3

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

[(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)] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The nesting number of a perfect matching.
This is the maximal number of chords in the standard representation of a perfect matching that mutually nest.

Code

def statistic(m):
    if not m:
        return 0
    return max(la[0] for la in to_oscillating(m) if la)

def to_oscillating(m):
    RuleRSK = GrowthDiagram.rules.RSK()
    m = PerfectMatching(m)
    n = m.size()
    filling = {(i-1, n-j): 1 for i, j in m.arcs()}
    shape = list(range(n-1,0,-1))
    return RuleRSK(filling, shape=shape).out_labels()

Created

Sep 10, 2020 at 21:17 by Martin Rubey

Updated

Feb 03, 2023 at 13:18 by Martin Rubey