On the parity of generalized partition functions III

TL;DR

This paper characterizes elements of a specific set where the associated partition function is even for all sufficiently large integers and provides an asymptotic estimate for the set's counting function.

Contribution

It extends previous results by explicitly identifying elements of a set related to partition functions and offers an asymptotic estimate for its size.

Findings

Identifies elements of set A with even partition functions for all n ≥ 6

Provides an asymptotic estimate for the counting function of set A

Improves upon earlier results by J.-L. Nicolas

Abstract

Improving on some results of J.-L. Nicolas \cite {Ndeb}, the elements of the set ${\cal A}={\cal A}(1+z+z^3+z^4+z^5)$, for which the partition function $p({\cal A},n)$ (i.e. the number of partitions of $n$ with parts in ${\cal A}$) is even for all $n\geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.