Number of unique pairings of k values.
You can get a formula (although maybe not very closed-form, you would definitely want a computer to do this for you) via that logic. First with one pairing, you have $n \choose 2$ possibilities. For 2 pairs, you have ${n\choose2} {n-2\choose 2}$, but notice that this has an order given to the two pairs, so we divide by 2!. For 3 pairs, you get $\frac1{3!}{n\choose2}{n-2\choose2}{n-4\choose2}$, and so on. So, you can write the answer as a sum.
From a quick calculation, you can find the values here: https://oeis.org/A001189 So for 13, there are 568503 pairings.