By doing some calculations on the generating function of matching polynomials of cycles I made the following interesting observation:


* For all positive integers $n>1$ and $k <n $, the number of matchings of size $k $ in $C\_{2n} $ is equal to the number of matchings of the same size in the disjoint union of two $C\_n $'s.


As mentioned, I have only an algebraic proof of this elementary result. Does anyone have a more illuminating proof?



Number of matchings of even cycles
There is in fact a topological(?) proof of this statement and the following generalization, essentially due to [Péter Csikvári](http://math.mit.edu/~csikvari/Schrijver_Gurvits_LMC_V6) (Section 4, Lemma 4.2).


We define a double cover (or ''2-lift'') $H$ of a graph $G$ as follows:
 consider $G$ as a topological space with CW structure, and $\pi:H\to G$ is
a topological double cover with CW structure induced from $G$.
Let $G\sqcup G$ denote the disconnected double cover of $G$. Let $m\_k(G)$ denote the number of $k$-edge matchings in $G$.



> 
> **Prop.** Let $G$ be a graph with no cycle of length smaller than $g$ (e.g. let $g$ be the girth of $G$) and let $H$ be a double cover of $G$. 
>  Then for any $k < g$,
>  $$m\_k(H) = m\_k(G\sqcup G) .$$
> 
> 
> 


*Proof*: Consider a $k$-matching $M\subset H$, and consider its image $\pi(M)$ in $G$. The image has vertices of valence at most 2, so $\pi(M)$ is a disjoint union of paths and cycles. But since $k<g$, there are no cycles. Finally, observe that any path in $G$ lifts to a matching in $H$ in exactly two ways, where $H$ is any double cover. Since $G\sqcup G$ is a double cover, the result follows. 




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Csikvári observed that when $G$ is bipartite, the same type of argument, now accounting for (even) cycles, implies that 
$$m\_k(H) \leq m\_k(G\sqcup G) $$
for any size matching $k$.
This is used to prove tight lower bounds on the number of matchings in a bipartite, $d$-regular graph.


