# Divisibility and Distribution of Partitions into Distinct Parts

@article{Lovejoy2001DivisibilityAD, title={Divisibility and Distribution of Partitions into Distinct Parts}, author={Jeremy Lovejoy}, journal={Advances in Mathematics}, year={2001}, volume={158}, pages={253-263} }

Abstract We study the generating function for Q ( n ), the number of partitions of a natural number n into distinct parts. Using the arithmetic properties of Fourier coefficients of integer weight modular forms, we prove several theorems on the divisibility and distribution of Q ( n ) modulo primes p ⩾5.

## 31 Citations

### The Number of Partitions into Distinct Parts Modulo Powers of 5

- Mathematics
- 2003

A relationship is established between the factorization of 24n + 1 and the 5‐divisibility of Q(n), where Q(n)is the number of partitions of n into distinct parts. As an application, an abundance of…

### The arithmetic of partitions into distinct parts

- Mathematics
- 2001

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n , and the number of such partitions is denoted by Q ( n ). If we adopt the…

### ARITHMETIC OF ℓ-REGULAR PARTITION FUNCTIONS

- Mathematics
- 2008

Let bl(n) denote the number of l-regular partitions of n, where l is prime and 3 ≤ l ≤ 23. In this paper we prove results on the distribution of bl(n) modulo m for any odd integer m > 1 with 3 ∤ m if…

### Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts

- Mathematics
- 2009

Let Q(n) denote the number of partitions of n into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the well-known fact that Q(n) is almost always divisible by 4.…

### Exact generating functions for the number of partitions into distinct parts

- MathematicsInternational Journal of Number Theory
- 2018

Let [Formula: see text] denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for [Formula: see text], [Formula: see text] and…

### Parity of sums of partition numbers and squares in arithmetic progressions

- Mathematics
- 2017

In this article, we explore the parity of sums of partition numbers at certain places in arithmetic progressions. In particular, we investigate pairs $$(a,b)\in \mathbb {N}^2$$(a,b)∈N2 for which if…

### The pa-Regular Partition Function Modulo pj☆

- Mathematics
- 2002

Abstract Let b l ( n ) denote the number of l-regular partitions of n , where l is a positive power of a prime p . We study in this paper the behavior of b l ( n ) modulo powers of p . In particular,…

### Distribution of the partitions of n in which no part appears exactly once

- Mathematics
- 2022

In this paper we study the function g ( n ), which denotes the number of partitions of n in which no part appears exactly once. We prove that for each prime m ≥ 5, there exist Ramanujan-type…

### 3-REGULAR PARTITIONS AND A MODULAR K3 SURFACE

- Mathematics
- 2004

In classical representation theory, k-regular partitions of n label irreducible kmodular representations of the symmetric group Sn when k is prime [8]. More recently, such partitions have been…

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