The classical Catalan numbers
$$ C_n = \frac{1}{n+1} \binom{2n}{n}, $$
well-known for their numerous combinatorial interpretations (the second volume of Stanley's *Enumerative Combinatorics* famously lists a total of 66), essentially arise as the even moments $E[X^{2n}] = \int x^{2n} d\mu$ of the Wigner semicircle distribution $d\mu$: this distribution depends on a parameter $R$, the radius of the semicircle, and in general one has $E[X^{2n}] = (R/2)^{2n} C_n$, so choosing $R=2$ gives precisely the Catalan numbers. (The odd moments are all zero.)

Now, associated to this distribution are a family of very classical orthogonal polynomials: the Chebyshev polynomials of the second kind, $U_n(x)$. (Up to a simple change of variables, $U_n(x)$ coincides with the matching polynomial of the path graph on $n$ vertices; for other interpretations and discussion see this blog post and this MO question.)

In their paper *Lectures on the topological recursion for Higgs bundles and quantum curves*, O. Dumitrescu and M. Mulase define a sequence of numbers $C_{g,n}(\mu_1,\ldots,\mu_n)$ which they call *generalized Catalan numbers*. The definition (found on p. 26) is as follows:

**Definition**: Let $\vec{\Gamma}_{g,n}(\mu_1,\ldots,\mu_n)$ denote the set of arrowed cell graphs drawn on a closed, connected, oriented surface of genus $g$ with $n$ labeled vertices of degrees $\mu_1,\ldots,\mu_n$. Then
$$ C_{g,n}(\mu_1,\ldots,\mu_n) := |\vec{\Gamma}_{g,n}(\mu_1,\ldots,\mu_n)|. $$

Below this, the authors justify the terminology by pointing out that $C_{0,1}(2m) = C_m = \frac{1}{m+1} \binom{2m}{m}$.

I don't know much about the story of multivariate orthogonal polynomials, and there are quite a few parameters here, but I can't help but wonder: might there perhaps be some multivariate analogue of the Wigner semicircle distribution, having these quantities $C_{g,n}(\mu_1,\ldots,\mu_n)$ as its (appropriately defined) moments – and with it, a corresponding family of orthogonal polynomials?

I also find it curious that the above paper discusses *Laplace transforms*, since the (bilateral) Laplace transform is exactly how one passes from a probability density function to its moment-generating function (as one learns in a first course on probability theory.) Admittedly, this is probably just a coincidence.