An iterative-bijective approach to asymmetric generalizations of Schur's theorem
Laure Velenik
TL;DR
The paper addresses generalizations of Schur-type partition theorems to colored overpartitions by developing an iterative-bijective proof framework. It proves that for any $k\ge1$, the infinite product $\frac{(-y_1 q;q)_\infty \cdots (-y_k q;q)_\infty}{(y_1 d q;q)_\infty}$ is the generating function for overpartitions whose parts come in $2^k-1$ colors, counting non-overlined parts via $d$ and the size via $q$. The main contribution is a bijective proof built by induction on $k$, unifying Lovejoy’s bijection for the $k=2$ case with the iterative approach of Corteel and Lovejoy, and providing a concrete combinatorial interpretation of a broad family of products. This work broadens the landscape of colored overpartition identities and suggests further iterated-bijective extensions and related $q$-series explorations with potential applications to symmetric and modular partition theorems.
Abstract
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: \[ \frac{(-y_1 q;q)_\infty \cdots (-y_k q;q)_\infty}{(y_1 d q;q)_\infty}. \] 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.