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Identifier

St000720: Perfect matchings ⟶ ℤ

Values

[(1,2)] => 1

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

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

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

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

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

[(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)] => 3

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

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

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

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

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

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

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

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

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

[(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)] => 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)] => 3

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

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

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

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

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

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

[(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)] => 3

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

[(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)] => 3

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

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

[(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)] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[(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)] => 2

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

[(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)] => 3

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

[(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)] => 3

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

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

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

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Description

The size of the largest partition in the oscillating tableau corresponding to the perfect matching.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.

Code

def statistic(m):
    return max((sum(1 for e in m if min(e) <= k and max(e) > k) for k in range(1, m.size())), default=0)

def statistic2(m):
    filling = {(min(e)-1, m.size()-max(e)): 1 for e in m}
    G = GrowthDiagram.rules.RSK()(filling, shape=[i for i in range(m.size()-1, 0, -1)])
    return max(sum(p) for p in G.out_labels()[1::2])

Created

Mar 26, 2017 at 00:21 by Martin Rubey

Updated

Feb 02, 2023 at 10:29 by Martin Rubey