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Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, Indalecio Ruiz

Abstract

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.

Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

Abstract

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by , , and working modulo and , respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.

Paper Structure

This paper contains 9 sections, 27 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. General Notation and Basic Results
  3. The Case of $\mathop{\mathrm{pd}}\nolimits_{2^\ell}(q) \, \operatorname{mod} 2$
  4. An Alternate Characterization of $\mathop{\mathrm{pd}}\nolimits_4(q) \, \operatorname{mod} 2$
  5. A Recurrence Relation for $\mathop{\mathrm{PD}}\nolimits_k(n) \, \operatorname{mod} 2$
  6. The Case of $\mathop{\mathrm{pd}}\nolimits_{2}(q) \, \operatorname{mod} 4$
  7. A Dissection Proof for Theorem \ref{['thm: pd4']}
  8. The Case of $\mathop{\mathrm{pd}}\nolimits_{3^\ell}(q) \, \operatorname{mod} 3$
  9. Open Problems

Key Result

Theorem 1.1

For all $n\geq0$,

Theorems & Definitions (57)

  • Theorem 1.1: ALL02
  • Theorem 1.2: dS20b
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 47 more