AlignedPairCount on PerfectMatching

Pairs of pairs (a, b), (c, d) with a < b < c < d.

On PerfectMatching

FindStat's St000719, its alignments. Any two pairs cross, nest or align, so the three counts sum to n choose 2.

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

Live

FindStat St000719

The number of alignments in a perfect matching.

Object	Value
[(1,2)]	0
[(1,2),(3,4)]	1
[(1,3),(2,4)]	0
[(1,4),(2,3)]	0
[(1,2),(3,4),(5,6)]	3
[(1,2),(3,5),(4,6)]	2
[(1,2),(3,6),(4,5)]	2
[(1,3),(2,4),(5,6)]	2

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000719 has the rest.