A variation on a theme of Sylvester -- a smoother road to Göllnitz's (Big) theorem
According to the celebrated theorem of \textit{H. Göllnitz} [J. Reine Angew. Math. 225, 154-190 (1967; Zbl 0166.00803)], the number of partitions of a positive integer \(n\) into distinct parts \(\equiv 2\), 4 or \(5\pmod 6\) is equal to the number of partitions of \(n\) in the form \(h_1+h_2+ \cdots +h_m\) such that \(h_m\neq 1\) or 3, and \(h_i-h_{i+1}\geq 6\) (for \(1\leq i<m)\) with strict inequality if \(h_i\equiv 0\), 1 or \(3\pmod 6\). \textit{K. Alladi}, \textit{G. E. Andrews} and \textit{B. Gordon} [J. Reine Angew. Math. 460, 165-188 (1995; Zbl 0810.11061)] obtained substantial generalizations and refinements of Göllnitz's theorem. In the paper under review the author gives a combinatorial proof of the following Theorem. The number of partitions of a positive integer \(n\) into distinct odd parts is equal to the number of partitions of \(n\) into parts \(\neq 2\) and differing by \(\geq 6\), where the inequality is strict whenever a part is even. The real interest in this Theorem comes from the fact that it has a three-parameter refinement which is much deeper. This refinement is stated as Theorem 2 in Section 4 and it is a reformulation of a three-parameter refinement of Göllnitz's theorem.
partitions with difference conditions and congruential restrictions
scientific article; zbMATH DE number 1321764