Some sieves for partition theory (Q1134176)

Adapted author's abstract: A combinatorial lemma due to \textit{J. Zolnowsky} [Discrete Math. 9, 293--298 (1974; Zbl 0292.10014)], is applied to partition theory in an ingenious way. For instance if \(E_{x,y}(n)\), resp. \(O_{x,y}(n)\), denotes the number of partitions of \(n\) into an even, resp. odd number of distinct parts congruent to \(0, \pm x \pmod y\), then \[ E_{x,y}(n)-O_{x,y}(n)=\begin{cases} (-1)^k &\text{ if } n=(k/2)[yk\pm(y-2x)] \\ 0 &\text{ otherwise}.\end{cases} \] Further a combinatorial proof of the Jacobi triple-product identity is given, based on the previous result. Some sieve formulas are proved which relate to the Euler pentagonal number theorem and the Rogers-Ramanujan identities and their generalizations. A formula is given for the number of partitions of \(n\) into parts not congruent to \(0, \pm x \pmod y\).