Number of crossing 1-factors in $K_{2n}$ Given a vertex set $\{1,\ldots,n\}$,
two edges $\{i,j\}$ and $\{k,\ell\}$ are crossing iff $i\lt k\lt j \lt \ell$.
I'm interested in the number of 1-factors in the complete graph over $2n$ vertices, such that each edge of the 1-factor crosses some other edge of the 1-factor.
Does anyone have any ideas about this?
Edit: This is closely related enumerating circle graphs.
Thank you
 A: OEIS knows about your sequence, which it calls Crossing matchings: linear chord diagrams with $2n$ nodes and $n$ arcs in which each arc crosses another arc.
Some highlights:
The sequence starts
$$ \begin{multline*} 1, 0, 1, 4, 31, 288, 3272, 43580, 666143, 11491696, \\ 220875237, 4681264432, 108475235444, 2728591657920, 74051386322580, \\ 2156865088819692, 67113404608820943, \\ 2221948578439255200, 77990056655776149179, \ldots \end{multline*} $$
The generating function $F$ is the solution of
$$ F' = \frac{-x^2F^3 + F - 1}{2x^3F^2 + 2x^2F}. $$
There is a reference: M. Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Adv. in Appl. Math., 30 (2003), 126-136.
There is a link: Alexander Stoimenow, On enumeration of chord diagrams and asymptotics of Vassiliev invariants, Chapter 3.

If you're wondering how I found the link: I just calculated the first 4 numbers and asked OEIS. I could also have guessed the name, though to be extra sure you need to calculate some numbers anyway. 
