Overpartitions and generating functions for generalized Frobenius partitions. (English) Zbl 1068.05005

Drmota, Michael (ed.) et al., Mathematics and computer science III. Algorithms, trees, combinatorics and probabilities. Proceedings of the international colloquium of mathematics and computer sciences, Vienna, September 13–17, 2004. Basel: Birkhäuser (ISBN 3-7643-7128-5/hbk). Trends in Mathematics, 15-24 (2004).
A generalized Frobenius partition of \(n\) is a pair \((\lambda,\mu)\) of partitions \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_m)\) and \(\mu=(\mu_1,\mu_2,\dots,\mu_m)\) such that \( m+\sum _{i=1} ^{m}(\lambda_i+\mu_i)=n\). Given two sets of (ordinary) partitions \(A\) and \(B\), let \(P_{A,B}(n)\) denote the number of generalized Frobenius partitions \((\lambda,\mu)\) of \(n\) in which \(\lambda\in A\) and \(\mu\in B\). The authors find several elegant product formulae for the generating function \( \sum _{n\geq0} ^{}P_{A,B}(n)\,q^n\) (and more refined generating functions), where \(A\) and \(B\) are special sets of overpartitions. Here, an overpartition is an ordinary partition where the first occurrence of a part may be overlined. As a sample result, I cite \[ \sum _{m,n\geq0} ^{}P_{O_k,O_k}(m,n)\,b^mq^n= \frac {(-bq;q)_\infty\,(-q/b;q)_\infty\,(q^k;q^k)_\infty} {(q;q)_\infty^2\,(-bq^k;q^k)_\infty\,(-q^k/b;q^k)_\infty}, \] where \(O_k\) denotes the set of overpartitions where the non-overlined parts occur less than \(k\) times, and where \((\alpha;q)_\infty= \prod _{i\geq1} ^{}(1-\alpha q^i)\). The proofs involve a tricky \(q\)-series lemma and classical identities for basic hypergeometric series. In some special cases, the authors also offer bijective proofs of their identities. This continues earlier work of the authors [Trans. Am. Math. Soc. 356, 1623–1635 (2004; Zbl 1040.11072)].
For the entire collection see [Zbl 1047.68003].


11P81 Elementary theory of partitions
05A17 Combinatorial aspects of partitions of integers
05A15 Exact enumeration problems, generating functions




Zbl 1040.11072