An iterative-bijective approach to asymmetric generalizations of Schur's theorem

In this paper, we present a new Rogers–Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur’s theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial interpretation of the following product, for any positive integer $k$, as a generating function for a class of overpartitions in which parts appear in $2^k-1$ colors: $$\frac{(-y_{1}q;q{)}_{\mathrm{\infty }}\cdots (-y_{k}q;q{)}_{\mathrm{\infty }}}{(y_{1}dq;q{)}_{\mathrm{\infty }}}\mathrm{.}$$ Our proof is bijective and unifies two earlier approaches: Lovejoy’s bijective proof of the asymmetrical Schur theorem and the iterative-bijective technique developed by Corteel and Lovejoy.