15 November 2007

Asymptotics of partition polynomials

I've previously mentioned Richard Stanley's page of interesting zeros of polynomials. For an example of this in action, you can see a couple papers which were justed posted to the arXiv, by Robert P. Boyer and William M. Y. Goh:

Partition Polynomials: Asymptotics and Zeros

Polynomials associated with Partitions: Their Asymptotics and Zeros

In the first paper, we take the polynomial which counts the partitions of n by number of parts; when n = 200 the zeroes of this polynomial are plotted at Richard Stanley's web site. What's surprising is that the zeroes are actually a good bit more subtle than they'd look from that picture; they come in three families, the third of which doesn't appear until n = 13,000.

The second paper makes a similar conjecture for the zeroes of the "rank polynomial" and "crank polynomial" of high degree, where partitions are counted by their rank (the difference between the largest part and the number of parts) or the crank (see p. 11 for definition). (In the rank case, at least, Boyer and Goh actually just count partitions with positive rank; by conjugation one can get those with negative rank.) The zeroes of the rank polynomial appear to lie approximately on the unit circle, and are spread out uniformly.

By the way, I've previously credited this idea of looking at the zeros of combinatorially defined polynomials to Stanley. Apparently it's actually due to Rota, who once said:
The one contribution of mine that I hope will be remembered has consisted in just pointing out that all sorts of problems of combinatorics can be viewed as problems of location of the zeros of certain polynomials and in giving these zeros a combinatorial interpretation. This is now called the critical problem. Over the years people have added examples of critical problems, though we still haven't gotten any idea of how to solve it in general. I'd like to see someone get such an idea before I die. The four-color conjecture-that with only four colors you can color every planar map so that no two adjacent regions have the same color-is one of these critical problems.

Rota was Stanley's thesis advisor.

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