Generalisations of the Andrews-Olsson partition identity and applications
The celebrated 1926 partition theorem of Schur is the statement that the number of partitions of \(n\) into distinct parts \(\equiv 1\) or \(2\pmod 3\) is equal to the number of partitions of \(n\) into parts differing by \(\geq 3\) and where consecutive multiples of 3 cannot occur as parts. In the 1960's, George Andrews obtained an extension of this result where the modulus 3 is replaced by an integer \(M\geq 3\) and the residue classes \(1, 2\pmod 3\) are replaced by \(r_i\pmod M\) where the \(r_i\) satisfy certain conditions. In 1991, \textit{G. E. Andrews} and \textit{J. B. Olsson} [J. Reine Angew. Math. 413, 198-212 (1991; Zbl 0704.20003)] obtained an important reformulation of this general result by using conjugation of Ferrers graphs. Also in 1991, the author [Eur. J. Comb. 12, No. 4, 271-276 (1991; Zbl 0743.05003)] gave a combinatorial proof of the Andrews-Olsson partition identity. In this paper the author obtains a further generalization by removing the conditions on the residues \(r_i\pmod M\) that Andrews and Olsson had. In doing so, the author obtains as a special case, a generalization of the Schur partition theorem due to the reviewer and \textit{B. Gordon} [Manuscr. Math. 79, No. 2, 113-126 (1993; Zbl 0799.11043)].
scientific article; zbMATH DE number 778520