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Lattice paths and Rogers--Ramanujan--Gordon type overpartitions

Diane Y. H. Shi

TL;DR

The paper builds a rigorous bridge between Rogers-Ramanujan-Gordon-type overpartitions and lattice paths with four unit steps, using a bijection based on Gordon marking to translate partition conditions into path constraints. It then extends Andrews' parity ideas to lattice paths, developing parity-controlled constructions that yield new generating functions for parity-restricted families of paths and their overpartition analogues. Through a sequence of bijective mappings and path operations, the authors derive lattice-path interpretations of key identities (including Chen–Sang–Shi forms) and establish GT-type generating functions linking peak and cluster parities. The results provide combinatorial proofs and new parity insights that deepen the connection between overpartitions and lattice-path combinatorics, with potential implications for further Rogers-Ramanujan–Gordon-type identities. The work thus offers a unified framework to study overpartition identities via lattice-path techniques and parity indexing.

Abstract

In this paper, we establish a connection between Rogers-Ramanujan-Gordon type overpartitions to lattice paths with four kinds of unitary steps. By establishing the bijective relationship between overpartitions and lattice paths, we demonstrate that the theorems provided by Chen, Sang and Shi can be formulated in the form of lattice paths. Subsequently, inspired by Andrews' work on parity in partition identities and its related implications, we impose parity constraints on lattice paths and present some novel discoveries. By leveraging the parity outcomes within lattice paths, we also revisit overpartitions to derive results pertaining to overpartitions with parity considerations.

Lattice paths and Rogers--Ramanujan--Gordon type overpartitions

TL;DR

The paper builds a rigorous bridge between Rogers-Ramanujan-Gordon-type overpartitions and lattice paths with four unit steps, using a bijection based on Gordon marking to translate partition conditions into path constraints. It then extends Andrews' parity ideas to lattice paths, developing parity-controlled constructions that yield new generating functions for parity-restricted families of paths and their overpartition analogues. Through a sequence of bijective mappings and path operations, the authors derive lattice-path interpretations of key identities (including Chen–Sang–Shi forms) and establish GT-type generating functions linking peak and cluster parities. The results provide combinatorial proofs and new parity insights that deepen the connection between overpartitions and lattice-path combinatorics, with potential implications for further Rogers-Ramanujan–Gordon-type identities. The work thus offers a unified framework to study overpartition identities via lattice-path techniques and parity indexing.

Abstract

In this paper, we establish a connection between Rogers-Ramanujan-Gordon type overpartitions to lattice paths with four kinds of unitary steps. By establishing the bijective relationship between overpartitions and lattice paths, we demonstrate that the theorems provided by Chen, Sang and Shi can be formulated in the form of lattice paths. Subsequently, inspired by Andrews' work on parity in partition identities and its related implications, we impose parity constraints on lattice paths and present some novel discoveries. By leveraging the parity outcomes within lattice paths, we also revisit overpartitions to derive results pertaining to overpartitions with parity considerations.
Paper Structure (7 sections, 21 theorems, 67 equations, 9 figures)

This paper contains 7 sections, 21 theorems, 67 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. The bijection between overpartition and lattice paths with four kinds of unitary steps
  4. The proof of Theorem \ref{['FD']}
  5. The proof of Theorem \ref{['thmH']} and \ref{['thmJ']}
  6. The proofs of Theorem \ref{['thmO']} and \ref{['thmoverO']}
  7. The proof of Theorem \ref{['thmGT']}

Key Result

Theorem 1.1

For $k\geq a\geq 1$, let $\overline{B}_{k,a}(n)$ denote the number of overpartitions of $n$ of the form $\lambda_1+\lambda_2+\cdots+\lambda_s$, such that $1$ can occur as a non-overlined part at most $a-1$ times, and where $\lambda_j-\lambda_{j+k-1}\geq1$ if $\lambda_j$ is overlined and $\lambda_j-\

Figures (9)

  • Figure 1: The two kinds of peaks.
  • Figure 2: The first dilation operations on lattice path of Type A.
  • Figure 3: The first dilation operations on lattice path of Type B.
  • Figure 4: The rules for increaseing the weight of a NSE peak with relative height one by one
  • Figure 5: The rules for increasing the weight of an NESE peak with relative height one by one
  • ...and 4 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 20 more