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Congruences for two-color partitions with odd smallest part

George E. Andrews, Mohamed El Bachraoui

Abstract

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$.

Congruences for two-color partitions with odd smallest part

Abstract

For a fixed positive integer , let denote the number of two-color partitions of with odd smallest part and restrictions on even parts, and let be its generating function. We show that and obtain congruences modulo and for when . Using -series methods we derive closed formulas for in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from .

Paper Structure

This paper contains 10 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. $q$-series background
  3. Proof of Theorem \ref{['thm cong1']}
  4. Proofs of Corollaries \ref{['cor odd-C1']}-\ref{['cor 9n+8']}
  5. Proof of Theorem \ref{['thm cong2']}
  6. Proof of Theorem \ref{['thm cong3']}
  7. Proof of Theorem \ref{['thm C1']}
  8. Proof of Theorem \ref{['thm C21']}
  9. Proof of Theorem \ref{['thm C31']}
  10. Remarks and conjectures

Key Result

Theorem 1

For any positive integer $n$, we have

Theorems & Definitions (26)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 16 more