Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell C. Hammon, Joel Henningsen, Henry Ickes, Jorge Marchena Menendez, Taylor Poe, Indalecio Ruiz, Edward L. Smith

Abstract

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

Abstract

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.

Paper Structure

This paper contains 11 sections, 9 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Initial Definitions and Statement of Main Theorem
  3. Combinatorial Identities
  4. Combinatorial Definitions
  5. Bijection between $B_{k,m}(n)$ and $C_{k,m}(n)$ for $m=0,1$
  6. Bijection between $A_k(n) \, \dot\cup \, A'_k(n)$ and $C_{k,0}(n) \, \dot\cup \, C_{k,1}(n)$
  7. Proof of Main Result
  8. Connections to Andrews, Merca
  9. The Generating Function for $a_k(n)$
  10. Recurrence Relations for $a_k(n)$
  11. Concluding Remarks

Key Result

Theorem 1.1

For any partition $\lambda = (\lambda_1^{m_1}, \lambda_2^{m_2}, \ldots, \lambda_\ell^{m_\ell})$, denote by $\gamma_{\mathcal{O}}(\lambda)$ the number of even bases $\lambda_i$, and by $\gamma_{\mathcal{D}}(\lambda)$ the number of repeated bases $\lambda_i$. Then, for $j \ge 0$, the number of partiti

Theorems & Definitions (26)

  • Theorem 1.1: P
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 16 more