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

Key Points

The study establishes a new identity for overpartitions, which extends Schur's theorem to a wider class of products.
A combinatorial interpretation of the identity is provided, using colors in a specific way related to positive integers.
The proof unifies previous approaches through a bijective method, enhancing the understanding of Schur's theorem.
The research shows that the generating function can effectively describe a new class of combinatorial structures.

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 2ᵏ - 1 colors: \ (-y₁ q;q) _ (-yₖ q;q) (ₘ䃑 ₃ ₐ;ₐ) _. \ 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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Cite This Study

Laure Velenik (Wed,) studied this question.

synapsesocial.com/papers/68e25378d6d66a53c247425f https://doi.org/https://doi.org/10.48550/arxiv.2510.00846

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