Stabilization of coefficients for partition polynomials

TL;DR

This paper investigates the stabilization behavior of various partition statistics, establishing bounds, exploring variants, and identifying limiting sequences through generating functions, with applications to Euler product forms and overpartition structures.

Contribution

It introduces a unified framework for understanding stabilization phenomena in partition statistics and provides explicit bounds and limiting sequences for numerous cases.

Findings

Partition statistics exhibit stabilization similar to p_k(n).

Explicit bounds for stabilization regions are derived.

Limiting sequences are identified via generating functions.

Abstract

We find that a wide variety of families of partition statistics stabilize in a fashion similar to $p_k(n)$, the number of partitions of n with k parts, which satisfies $p_k(n) = p_{k+1}(n + 1), k \geq n/2$. We bound the regions of stabilization, discuss variants on the phenomenon, and give the limiting sequence in many cases as the coefficients of a single-variable generating function. Examples include many statistics that have an Euler product form, partitions with prescribed subsums, and plane overpartitions.