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

Laure Velenik

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arXiv · 2510.00846

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

Laure Velenik

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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.

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Laure Velenik. 2025-10-01. An iterative-bijective approach to asymmetric generalizations of Schur's theorem. https://arxiv.org/abs/2510.00846

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