It is a well-known exercise that $C\_n = \chi\_{(n,n)}(1)=\chi\_{(n,n)}^{1^n}$ where $C\_n$ is the $n$th Catalan number and $\chi\_{(n,n)}^{1^n}$ is the character of the irrep $(n,n)$ on conjugacy class $1^n = (1,1,\cdots,1)$.


A related identity that I stumbled across for fixed-point-free involutions is that $\chi\_{(n,n)}^{2^n} = \pm \binom{n}{\lfloor n/2 \rfloor}$. 


I haven't been able to find this stated as an exercise in a textbook or online manuscript, and it seems too obvious to have been overlooked. If you have seen this identity before and recall the reference, I would be happy to know. 



A Combinatorial Identity Involving Characters of $S_n$ (Reference Request?)
This is a special case of Corollary 2.2 in S. Fomin and N. Lulov, [On the number of rim-hook tableaux](http://www.math.lsa.umich.edu/~fomin/Papers/lulov.ps), taking $r=2$ and $\lambda = (n,n)$. It's probably easier to use the formula in the proof than the corollary itself.


