On the subpartitions of the ordinary partitions (Q618855)

A subpartition with gap \(d\) of an ordinary partition \(a_1\geq a_2\geq\cdots\geq a_{\ell}\) is defined as the longest sequence satisfying \(a_1>a_2>\cdots> a_s\) and \(a_s>a_{s+1}\), where \(a_i-a_j\geq d\) for all \(i<j\leq s\). This definition is a generalization of the Rogers-Ramanujan subpartition introduced by {L. W. Kolitsch} in [Ramanujan J. 16, No. 2, 163--167 (2008; Zbl 1170.11036)], which corresponds to the case \(d=2\) in the definition above. The author investigates several properties of subpartitions with gap \(d\) and, as an application using the subpartition with gap 1, presents a combinatorial proof of the identities \[ \frac{1}{(q)^2_{\infty}}\sum_{n=0}^{\infty}(-1)^nq^{(n^2+n)/2}=\sum_{n=0}^{\infty}\frac{q^n}{(q)^2_n} \] and \[ \frac{1}{(q)^2_{\infty}}\left(1+2\sum_{n=1}^{\infty}(-1)^nq^{(n^2+n)/2}\right)=\sum_{n=0}^{\infty}\frac{q^{2n}}{(q)^2_n}, \] which are entries in Ramanujan's lost notebook [\textit{G. E. Andrews} and \textit{B. C. Berndt}, Ramanujan's lost notebook. Part II. New York, Springer (2009; Zbl 1180.11001), p. 19, Entries 1.4.10 and 1.4.11,8].