Thanks for this reformulation. Is it possible to restate it in purely enumerative terms?

There are a vast number of papers on special cases, one example being Goupil's paper at sciencedirect.com/science/article/pii/0012365X9090054L. I wouldn't be surprised if the general problem is #P-complete. Maybe this is already known. See also inria.hal.science/inria-00098749/document.

Essentially a repeat of mathoverflow.net/questions/62088.

If your analytic curve satisfies a linear differential equation with polynomial coefficients, then the book D-Finite Functions by Manuel Kauers contains much information. See link.springer.com/book/10.1007/978-3-031-34652-1.

For a reference, see the paragraph after Theorem 1.1 of R. M. Erdahl, Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra (sciencedirect.com/science/article/pii/…).

An asymptotic estimate for $p(n,k)$ was obtained by G. Szekeres, Some asymptotic formulae in the theory of partitions (II), Quart. J. Math. Oxford 4 (2), 96--111, that implies the unimodality of the sequence $p(n,1), p(n,2), \dots, p(n,n)$. Does this estimate or some refinement of it imply log-concavity?

For the number of irreps of S_n whose dimension is relatively prime to some fixed prime $p$, see Enumerative Combinatorics, vol. 2, Exercise 7.15 (due to I. G. Macdonald). For a variant, not well understood, see Supplementary Exercise 32 in the second edition of the reference above.