A bijective proof of a false theta function identity from Ramanujan's lost notebook

In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n{q}^{n(n+1)\mathrm{/}2}\mathrm{.}$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtained a combinatorial interpretation of another one of the identities.