Combinatorial polynomials from general diagram fillings? There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams.
There are also composition shaped diagrams that are responsible for Demazure characters and some other non-symmetric polynomials, such as the non-symmetric Macdonald polynomials.
However, I am not aware of any family of polynomials that are defined on more general shapes of diagrams (well, except perhaps the shifted tableaux, that define $P$-Schur polynomials).
Thus, what are some families of polynomials (or families of fillings), defined on more exotically shaped diagrams?
 A: To any poset $P$ and any labeling $\omega$ of $P$ we can associate the ``polynomial'' (actually, formal power series) $K(P,\omega) := \sum_{\sigma \in A^r(P,\omega)} x^{\sigma}$. Here $A^r(P,\omega)$ is the set of all reverse $(P,\omega)$-partitions (i.e., fillings of the poset $P$ with entries $\mathbb{N}$ obeying certain inequalities depending on $\omega$ and the order structure) and we use the notation $x^f := \prod_{i \geq 1} x_i^{\#f^{-1}(i)}$. Then $K(P,\omega)$ is always a quasisymmetric function. See section 7.19 of Stanley's Enumerative Combinatorics vol 2 for more details.
When $P$ is the poset of a skew diagram and $\omega$ is a compatible ``Schur labelling'' we can say more: in this case $K_{(P,\omega)}$ is symmetric. In fact, I believe it is a conjecture of Stanley from circa 1970 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 $P_{\lambda/\mu}$.
