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 $S_n$.

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...?): $\mathcal H_n(g,k) = \sum_{\nu \in P(n)} \left(\prod_{i=1} ^kccl(\mu^i) \frac{\chi_\nu(\mu^i)}{\chi_\nu(1)}\right) \left( \frac{\chi_\nu(1)}{n!} \right)^{22g}$. Here $\nu \in P(n)$ indexes the irreducible representations $\chi_\nu$ of $S_n$, and the $\mu^i$ are identifed with conjugacy classes in $S_n$. 


As a somewhat trivial example, the sign of a permutation is a character of $S_n$, and the sign function appears in many contexts. For instance, the sign function is used when calculating the determinant of a matrix by summing over permutations in $S_n$. Replacing the sign character in this formula with another character of $S_n$ gives the definition of the immanant of a matrix. 

