Partitions with rounded occurrences and attached parts
Define the series \(S_{k,i}(a,x,q)\) as \[ \begin{multlined} S_{k,i}(a,x,q) = \frac{1}{(xq)_\infty}\sum_{n\geq 0}\frac{a^n x^{(k-1)n}q^{(k-1)n^2+(k-i+1)n} (x^2q^2, 1/a; q^2)_n}{(q^2,ax^2q^2; q^2)_n}\\ \left(1+\frac{ax^iq^{(2n+1)i-2n}(1-q^{2n}/a)}{1-ax^2q^{2n+2}}\right),\end{multlined} \] and write \(S_{k,i}(a,x,q)=\sum_{t,m,n} s_{k,i}(t,m,n) a^t x^m q^n\). The author gives a combinatorial interpretation of the coefficients \(s_{k,i}\) as the number of pairs of partitions \((\lambda, \mu)\) of \(n\) satisfying certain conditions too complicated to be stated here. Although the definition of these restrictions is highly involved, the result is surprisingly useful, as the author demonstrates by giving a new proof of a theorem by \textit{D. M. Bressoud} [``A generalization of the Rogers-Ramanujan identities for all moduli,'' J. Comb. Theory, Ser. A 27, 64--68 (1979; Zbl 0416.10009)] and proving a new identity for the number of certain pairs of overpartitions. The proof uses standard computations involving hypergeometric series.
scientific article; zbMATH DE number 5837834
scientific article