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"Overpartitionized" Rogers--Ramanujan type identities

Abdulaziz Alanazi, Augustine O. Munagi, Andrew V. Sills

TL;DR

The paper develops overpartition interpretations for classical Rogers-Ramanujan type identities, showing that important partition counts (including the RR, GG, and LG families) have natural overpartition realizations with precise parity and size constraints. It provides generating-function proofs, explicit bijections, and Stembridge-type interpretations, unifying classical identities with overpartition combinatorics. A key contribution is a parametric Lebesgue identity framework and its specializations that yield broad families of overpartition counts and alternative Frobenius–diagram interpretations, including self-conjugate and almost-self-conjugate structures. These results deepen connections between $q$-series identities and combinatorial partition theory, offering new tools for bijective and analytic study of overpartitions in Rogers– Ramanujan type settings.

Abstract

Many classical $q$-series identities, such as the Rogers--Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical $q$-series to yield natural combinatorial interpretations in terms of overpartitions. Bijective proofs are supplied as well.

"Overpartitionized" Rogers--Ramanujan type identities

TL;DR

The paper develops overpartition interpretations for classical Rogers-Ramanujan type identities, showing that important partition counts (including the RR, GG, and LG families) have natural overpartition realizations with precise parity and size constraints. It provides generating-function proofs, explicit bijections, and Stembridge-type interpretations, unifying classical identities with overpartition combinatorics. A key contribution is a parametric Lebesgue identity framework and its specializations that yield broad families of overpartition counts and alternative Frobenius–diagram interpretations, including self-conjugate and almost-self-conjugate structures. These results deepen connections between -series identities and combinatorial partition theory, offering new tools for bijective and analytic study of overpartitions in Rogers– Ramanujan type settings.

Abstract

Many classical -series identities, such as the Rogers--Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical -series to yield natural combinatorial interpretations in terms of overpartitions. Bijective proofs are supplied as well.

Paper Structure

This paper contains 9 sections, 12 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Euler's Distinct Parts Partitions
  3. The Rogers--Ramanujan Identities
  4. The Göllnitz--Gordon Identities
  5. The Little Göllnitz Identities
  6. General Case
  7. Stembridge-type Interpretations
  8. Definitions and Stembridge results
  9. Almost Self-Conjugate Partitions

Key Result

Proposition 2.1

Let $D(n)$ denote the number of partitions of $n$ into distinct parts and let $\overline{D_k}(n)$, where $k\geq 1$, denote the number of overpartitions of $n$ in which the non-overlined parts are distinct and at least $k$ and the overlined parts are at most $k-1$. Then for any fixed positive integer $k$.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Bijective Proof of Theorem \ref{['D']}
  • Theorem 3.1
  • proof : Generating function proof
  • proof : Bijective proof
  • ...and 23 more