Speed dating model
I think the question can be rephrased more formally as: Given a set $S$ of $n$ elements, what is the shortest sequence $C_i$ of sets of unordered pairs in $S$ such that each unordered pair occurs in exactly one $C_i$ and no pairs in a given $C_i$ "overlap"? Imagine a long table with a seat at one end and $\frac$ seats along each long side. After each round, each person moves one seat clockwise. This gets us N-1 rounds in the even case, which is optimal. I run gay speed dating events and have the seating charts for 12 participants up to 22.You should be able to convince yourself that each person meets each other after N rounds, but you can't do better as each person needs to meet N-1 others and has to sit out once. It's a little complex, but essentially you split the room into 2 parts, and then have 1/2 the room meet the other 1/2.
Also, we find that women exhibit a preference for men who grew up in affluent neighborhoods.
Try to phrase the problem in a more mathematical way.
Are there lesbians in this problem or is this the "Gay male speed dating problem"?
I.e.: could the gentlemen circle a rectangular table in a clockwise fashion, then rearrange themselves and continue in another fashion such that given any number of men, every man would be paired with every other man in the smallest number of iterations and without pairing two men together twice.
I'm not sure exactly what you mean by "counting combinations." Also, you need to keep cultural considerations in mind: probably not everyone who reads this will know what speed dating is.