Sets with even partition functions and 2-adic integers
For a set \(\mathcal A\) of positive integers let \(p(\mathcal A,n)\) denote the number of partitions of \(n\) into parts from \(\mathcal A\). \textit{J.-L.~Nicolas, I.~Z.~Ruzsa} and \textit{A.~Sárközy} [J. Number Theory 73, 292--317 (1998; Zbl 0091.11050)] proved that if \(\emptyset\neq \mathcal B\subseteq \{1,2,\dots,N\}\) then there exists a unique \(\mathcal A\) such that \(\mathcal A\cap \{1,\dots,N\}=\mathcal B\) and \(p(\mathcal A,n)\) is even for \(n>N\). In other words, for a nonconstant polynomial \(P\in GF(2)[z]\) with \(P(0)=1\), \(\exists !\) \(\mathcal A=\mathcal A(P)\) with the property \(\sum_{n\geq 0}p(\mathcal A,n)z^n\equiv P(z)\pmod 2\). For an odd prime \(p\), let \(P\) be some irreducible polynomial of order \(p\) (i.e., \(p\) is the smallest positive integer such that \(P(z)\) divides \(1+z^p\) in \(GF(2)[z]\)). In the paper under review the author proves that the elements of \(\mathcal A(P)\) of the form \(2^k m\) with an odd \(m\) are determined by the 2-adic expansion of some root of a resultant polynomial \(R_m(y)\in Z[y]\). E.g., \(R_1(y)=\text{res}_z\big(\frac{1+z^p}{1+z},\;y+\sum^{s-1}_{j=0}z^{\overline{2^j}}\big)\), where \(s\) is the order of 2 mod \(p\) and \(\overline{2^j}\) is the remainder of \(2^j\) mod \(p\). This extends a result of \textit{F.~Ben~saïd} and \textit{J.-L.~Nicolas} [Sémin. Lothar. Comb. 46, 25p. (2002; Zbl 1042.11008)] to all primes \(p\). In Section 4, the author makes the polynomial \(R_m(y)\) explicit in the case \(s=(p-1)/2\).
scientific article; zbMATH DE number 5368576