Partition Polynomials: Asymptotics and Zeros

Details Partition Polynomials: Asymptotics and Zeros Partition Polynomials: Asymptotics and Zeros

2007-11-09

arxiv.org/abs/0711.1373v1

Mathematics

Combinatorics

Scientific paper

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.

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