Ben Saïd, F.; Nicolas, J.-L.

Even partition functions

Sémin. Lothar. Comb. 46, 25 p. (2002)


Summary: Let $A$ be a set of positive integers. Let us denote by $p(A,n)$ the number of partitions of $n$ with parts in $A$. While the study of the parity of the classical partition function $p(N,n)$ (where N is the set of positive integers) is a deep and difficult problem, it is easy to construct a set $A$ for which $p(A,n)$ is even for $n$ large enough: as explained in a paper of I.Z. Ruzsa, A. Sárközy and J.-L. Nicolas published in 1998 in the Journal of Number Theory, if $B$ is a subset of ${1,2,\dots ,N}$, there is a unique set $A=A_{0}(B,N)$ such that the intersection of $A$ and ${1,2,\dots ,N}$ is equal to $B$ and $p(A,n)$ is even for $nN$. In this paper we recall some properties of the sets $A_{0}(B,N)$, we describe the factorization into primes of the elements of the set $A_{0}({1,2,3},3)$, and prove that the number of elements of this set up to $x$ is asymptotically equivalent to c x / $(\log x)^{3/4}$, where $c=0.937\dots $.