Polynomials with interesting sets of zeros

Polynomials with interesting sets of zeros, from Richard Stanley.

There are certain polynomials which are naturally defined in terms of combinatorial objects, which have interesting sets of zeros. For example, consider the polynomials

F₁(x) = x
F₂(x) = x + x²
F₃(x) = x + x² + x³
F₄(x) = x + 2x² + x³ + x⁴
F₅(x) = x + 2x² + 2x³ + x⁴ + x⁵

where the x^k coefficient of F_n(x) is the number of partitions of n into k parts, that is, the number of ways to write n as a sum of k integers where order doesn't matter. For example, the partitions of 5 are

5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1

which have 1, 2, 2, 3, 3, 4, and 5 parts respectively; that's where F₅ comes from. It turns out that if you take F_n for large n and plot its zeros in the complex plane, you get a nice-looking figure; see this PDF file. This is true of other combinatorially defined families of polynomials as well. The zeroes of this particular set of polynomials are given in some recent work of Robert Boyer and William Goh. (Boyer gave a talk today at Temple which I went to, hence why this is on my mind lately; I'd actually heard about this phenomenon when I took a class from Stanley as an undergrad.)

At least in the case of the partitions, it seems noteworthy that if we sum yⁿ F_n(x) over all n, getting the generating function of all partitions by size and number of parts, we have something nice, namely

and a lot of the other sets of polynomials described by Stanley can probably be described similarly. (Boyer and Goh allude to this at the end of their expository paper.) I don't know enough analytic number theory to say something intelligent about this, though; the proof that the pictures keep looking "like that" (in the case of partition) apparently uses some very heavy machinery.