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EarlyCloserPairCount on PerfectMatching ​

Pairs whose larger point is at most n + 1, for a matching of 2n points.

On PerfectMatching

Called as CombinatorialStat(x, "EarlyCloserPairCount"). The head's page has its examples.

Live

FindStat St001044

The number of pairs whose larger element is at most one more than half the size of the perfect matching.

ObjectValue
[(1,2)]1
[(1,2),(3,4)]1
[(1,3),(2,4)]1
[(1,4),(2,3)]1
[(1,2),(3,4),(5,6)]2
[(1,2),(3,5),(4,6)]1
[(1,2),(3,6),(4,5)]1
[(1,3),(2,4),(5,6)]2

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001044 has the rest.