Combinatorial proof of a partial theta function identity of Warnaar (Q2816447)

This paper provides a combinatorial proof for the following partial theta function identity due to \textit{S. O. Warnaar} [Proc. Lond. Math. Soc. (3) 87, No. 2, 363--395 (2003; Zbl 1089.05009)]: NEWLINE\[NEWLINE\sum_{n=0}^{\infty} (-1)^n a^n q^{n^2+n} = \sum_{n=0}^{\infty} \frac{(q;q^2)_n(aq;q^2)_n(aq)^n}{(-aq;q)_{2n+1}},NEWLINE\]NEWLINE where \((a;q)_n= \prod_{j=0}^{n-1}(1-aq^j)\) as usual. This is achieved by constructing a sign-reversing bijection on a suitably defined set of pairs consisting of an overpartition and a sequence of numbers.NEWLINENEWLINEMoreover, an Euler-like theorem for a specific kind of unimodal sequences is derived from the identity.