Cyclic Sieving of Matchings

The cyclic sieving phenomenon (CSP) was introduced by Reiner, Stanton, and White to study combinatorial structures with actions of cyclic groups. The crucial step is to find a polynomial, for example a q-analog, that satisfies the CSP conditions for an action. This polynomial will give us a lot of information about the symmetry and structure of the set under the action. In this paper, we study the cyclic sieving phenomenon of the cyclic group $C_{2n}$ acting on $P_{n,k}$, which is the set of matchings of $2n$ points on a circle with $k$ crossings. The noncrossing matchings ($k=0$) was recently studied as a Catalan object. In this paper, we study more general cases, the matchings with more number of crossings. We prove that there exists $q$-analog polynomials $f_{n,k}(q)$ such that $(P_{n,k},f_{n,k},C_{2n})$ exhibits the cyclic sieving phenomenon for $k=1,2,3$. In the proof, we also introduce an efficient representation of the elements in $P_{n,k}$, which helps us to understand the symmetrical structure of the set.