Number of possible pair sets
let's say we have k numbers from 1 to k and k is even. We want to know how many sets of pairs we are able to create with them. By a "pair set" we understand such a set that covers pairs created from all the k numbers given. (x, y) and (y, x) are treated as the same pair.
If k=2 then the answer is 1 since we can create only one pair which is (1,2).
If k=4 the answer is 3 since we can create 3 sets: ((1,2) and (3,4)) OR ((1,3) and (2,4)) OR ((1,4) and (2,3)).
If k=6 the answer is 15 since we can create 15 sets: ((1,2) and (3,4) and (5,6)) OR ((1,2) and (3,5) and (4,6)) OR ((1,2) and (3,6) and (4,5)) and so on.
I'm seeking a formula which would help me to calculate how many pair sets I am able to create with a given k numbers. Is there any?