Here's another classic from algebraic combinatorics.

In his PhD thesis "Ordered Structures and Partitions", Stanley introduces the $(P,\omega)$-partition generating function of a labeled poset $(P,\omega)$. This is defined to be $K(P,\omega) := \sum_{\sigma \in A^r(P,\omega)} x^{\sigma}$, where $A^r(P,\omega)$ is the set of all reverse $(P,\omega)$-partitions (i.e., fillings of the poset $P$ with entries in $\mathbb{N}$ obeying certain inequalities depending on $\omega$ and the order structure of $P$), and where we use the notation $x^f := \prod_{i \geq 1} x_i^{\#f^{-1}(i)}$. Stanley shows that $K(P,\omega)$ is always a quasisymmetric function. When $P$ is the poset of a skew Young diagram and $\omega$ is a compatible ``Schur labelling'' then $K_{(P,\omega)}$ is in fact a symmetric function (namely, a skew Schur funciton). Stanley's conjecture is that $K_{(P,\omega)}$ is a symmetric function if and only if $(P,\omega)$ is isomorphic to $(P_{\lambda/\mu},w)$, where $\lambda/\mu$ is a skew-shape and $w$ is a Schur labelling of $P_{\lambda/\mu}$. There has been some progress on this conjecture due to Malvenuto; I don't know the exact status of what has been proved.