What are the places where character table values of $S\_n$ occurs naturally? one such an example is when we write power sum symmetric function of order n in terms of Schur function of order n the coefficient of each monomial will be the character value of the respective conjugacy class and respective irreducible representation in the character table of $S\_n$. 



Character table of Sn
My favourite place where the character table of $S\_n$ occurs naturally is in the enumeration of branched covers of Riemann surfaces (Hurwitz numbers).


In more detail, let $\Sigma$ be a closed Riemann surface of $g$ with $k$ marked points $x\_1, \ldots, x\_k$, and fix an integer $n\geq 1$. For each marked point $x\_i$ pick a partition $\mu^i$ of $n$.


We define $\mathcal H\_n(g,\mu\_1 , \ldots ,\mu\_k)$ to be the sum of $1/|Aut(f)|$ for each isomorphism class of branched covers $f: \widetilde{\Sigma} \to \Sigma$ which are ramified precisely at $x\_i$ with ramification type $\mu^i$.


Then there is a formula due to Hurwitz (or perhaps Frobenius, or Burnside, depending on which version...?):


$\mathcal H\_n(g,k) = \sum\_{\nu \in P(n)} \left(\prod\_{i=1} ^k|ccl(\mu^i)| \frac{\chi\_\nu(\mu^i)}{\chi\_\nu(1)}\right) \left( \frac{\chi\_\nu(1)}{n!} \right)^{2-2g}$.


Here $\nu \in P(n)$ indexes the irreducible representations $\chi\_\nu$ of $S\_n$, and the $\mu^i$ are identifed with conjugacy classes in $S\_n$.


