Congruences for an analogue of Lin's partition function

TL;DR

This work introduces $B(n)$, an analogue of Lin's restricted partition function, with generating function $\sum_{n=0}^\infty B(n) q^n = \dfrac{f_2^4}{f_1^2 f_4^3}$ that counts triples under distinct-odd-part and multiples-of-4 constraints. Employing elementary $q$-series techniques, $\$-dissections, theta functions, and eta-quotients, it proves Ramanujan-type congruences modulo $2$, $3$, $5$, $7$, and $9$ for $B(n)$ and for several finite-sum expressions. The paper provides exact generating functions for subsequences like $B(3n+2)$ and $B(3n+1)$ modulo $3$ and $9$, and establishes congruences for certain weighted sums, expanding the landscape of partition congruences. It also demonstrates a computational modular approach via Radu's algorithm to obtain a mod $7$ generating function for $B(7n+2)$, highlighting the interplay between dissections, theta functions, and modular-function methods.

Abstract

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ and $π_2$ comprise distinct odd parts and $π_3$ consists of parts divisible by $4$. With the help of elementary $q$-series techniques and modular functions, we establish Ramanujan-type congruences modulo $2,3,5,7$, and $9$ for certain sums involving $B(n)$.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm13']}