The uncrossing partially ordered set Pn is defined on the set of matchings on 2n points on a circle represented with wires. The order relation is τ' ≤ τ in Pn if and only if 0 is obtained by resolving a crossing of τ. I identify elements in Pn with affine permutations of type (0, 2 n). Using this identidication, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset Pn. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset Pn. I also obtain a recursion for the ab-indices of intervals in the poset Pˆn, the poset P n with a unique minimum 0ˆ adjoined. Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the q-Catalan polynomial. Bowling-Liang presented similar results on the set of k-crossing matchings for 1 ≤ k ≤ 3. In this dissertation, I focus on the set of all matchings on 2 n := {1; 2, . . . ,2n}. I find the number of matchings fixed by 2π/d rotations for d|2 n. I then find the polynomial Xn( q) such that the set of matchings together with X n(q) and the cyclic group of order 2n exhibits the CSP.