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Lattice paths and the Rogers--Ramanujan--Gordon type theorems with parity considerations

Diane Y. H. Shi

Abstract

Andrews imposed parity restrictions on the Rogers-Ramanujan-Gordon type partitions, yielding fruitful results. These results were later, advanced by Kurşungöz, Kim, and Yee. In this paper, we construct a bijection between lattice paths with three kinds of unitary steps and the Rogers--Ramanujan--Gordon type partitions. We give some results involving parity considerations on lattice paths by this bijection which are Andrews' partition results counterpart. We also give some new results on lattice paths.

Lattice paths and the Rogers--Ramanujan--Gordon type theorems with parity considerations

Abstract

Andrews imposed parity restrictions on the Rogers-Ramanujan-Gordon type partitions, yielding fruitful results. These results were later, advanced by Kurşungöz, Kim, and Yee. In this paper, we construct a bijection between lattice paths with three kinds of unitary steps and the Rogers--Ramanujan--Gordon type partitions. We give some results involving parity considerations on lattice paths by this bijection which are Andrews' partition results counterpart. We also give some new results on lattice paths.
Paper Structure (4 sections, 15 theorems, 45 equations)

This paper contains 4 sections, 15 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Combinatorial interpretation of Andrews' function
  3. The bijection between Gordon marking of a partition and a lattice path
  4. The proofs of theorems

Key Result

Theorem 1.1

For $k \geq a\geq 1$, let $B_{k,a}(n)$ denote the number of partitions of $n$ of the form $\lambda_1+\lambda_2+\cdots+\lambda_s$, where $\lambda_j\geq \lambda_{j+1}$, $\lambda_j-\lambda_{j+2}\geq 2$ and at most $a-1$ of the $\lambda_j$ are equal to $1$. Let $A_{k,a}(n)$ denote the number of partitio

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: Andrews And10, Kurşungöz kur10, Kim and Yee kim13
  • Theorem 1.8: Andrews And10, Kurşungöz kur10, Kim and Yee kim13
  • Definition 1.9
  • Theorem 1.10
  • ...and 13 more