On the arithmetic properties of partitions into parts simultaneously $4$-regular and $9$-distinct

Technical Summary: Arithmetic Properties of Partitions into Parts Simultaneously 4-Regular and 9-Distinct

Problem Statement
This paper investigates the arithmetic properties of the partition function $RD(\ell, t)(n)$, which counts the number of partitions of an integer $n$ into parts that are simultaneously $\ell$-regular (no part is divisible by $\ell$) and $t$-distinct (each part appears fewer than $t$ times). While previous work by the authors and others has explored various pairs of $(\ell, t)$, this study focuses specifically on the case $(\ell, t) = (4, 9)$. The function $RD(4, 9)(n)$ corresponds to the sequence A187020 in the Online Encyclopedia of Integer Sequences (shifted by one term). The primary objective is to establish infinite families of congruences and Ramanujan-like congruences for $RD(4, 9)(n)$ modulo 4, 6, 12, and 24.

Methodology
The authors employ the theory of $q$-series and generating functions to derive their results. The core methodology involves:

1. Generating Function Dissections: Utilizing known dissection formulas for theta functions and products of $f_k = (q^k; q^k)_\infty$. Specifically, the paper relies on 2-dissections (Lemma 2.1), 5-dissections (Lemma 2.2), and general prime dissections for theta functions (Lemmas 2.3 and 2.4) attributed to Ramanujan, Cui, Gu, and others.
2. Modular Reductions: Applying the congruence $f_{p^k} \equiv f_{p^{k-1}} \pmod{p^k}$ (Lemma 2.5) to simplify generating functions modulo specific integers.
3. Coefficient Extraction: Systematically extracting terms with specific powers of $q$ (e.g., $q^{pn+i}$) from the dissection formulas to isolate subsequences of the partition function.
4. Quadratic Residue Analysis: Analyzing congruences involving quadratic residues (e.g., $(6k+1)^2 + (6m+1)^2 \equiv 0 \pmod p$) to demonstrate that certain coefficients vanish for specific primes $p$ satisfying conditions like $p \equiv 3 \pmod 4$ or $p \equiv 5 \pmod 6$.
5. Mathematical Induction: Using the vanishing of specific coefficients to establish infinite families of congruences via induction on the exponent $\alpha$.

Key Contributions and Results

• Modulo 4:
  The paper establishes an infinite family of congruences for primes $p \equiv 3 \pmod 4$. For any $\alpha \ge 0$, $n \ge 0$, and $1 \le i \le p-1$:
  $$RD(4, 9)(12p^{2\alpha+1}(pn + i) + p^{2\alpha+2} + 1) \equiv 0 \pmod 4.$$
  This result is derived by relating $RD(4, 9)(12n+2)$ to the coefficients of $f_1^2$ and utilizing the non-existence of non-trivial solutions to specific quadratic congruences modulo $p$.

• Modulo 6:
  An infinite family of congruences is proven for powers of 5. For all $\alpha \ge 0$ and $n \ge 0$:
  $$RD(4, 9)(6 \cdot 5^{2\alpha+2}n + 6 \cdot 5^{2\alpha+1}i + 5^{2\alpha+2} + 1) \equiv 0 \pmod 6,$$
  where $i \in \{1, 2, 3, 4\}$. This is achieved by dissecting the generating function for $RD(4, 9)(6n+2)$ and applying the 5-dissection of Ramanujan's function $f_1$.

• Modulo 12:
  For primes $p \equiv 5 \pmod 6$, the authors prove that for all $\alpha \ge 0$, $n \ge 0$, and $1 \le i \le p-1$:
  $$RD(4, 9)(6p^{2\alpha+1}(pn + i) + 3p^{2\alpha+2} + 1) \equiv 0 \pmod{12}.$$
  The proof involves analyzing the product of theta functions $\psi(q)\psi(q^3)$ and showing that coefficients vanish when the exponent satisfies specific modular conditions related to quadratic non-residues of $-3$.

• Modulo 24 (Ramanujan-like Congruences):
  The paper establishes specific Ramanujan-like congruences that do not depend on an infinite family parameter but hold for all $n \ge 0$:
  $$RD(4, 9)(24n + 23) \equiv 0 \pmod{24},$$
  $$RD(4, 9)(48n + 29) \equiv 0 \pmod{24},$$
  $$RD(4, 9)(96n + 89) \equiv 0 \pmod{24}.$$
  These are derived through a rigorous process of extracting even and odd powers from the generating function of $RD(4, 9)(6n+5)$ and applying 2-dissection formulas.

Significance and Scope
The paper claims to provide a "thorough investigation" of the arithmetic behavior of $RD(4, 9)(n)$. Its significance lies in extending the study of partitions that are simultaneously regular and distinct, a topic initiated by Keith and previously explored by the authors. By establishing these congruences, the authors contribute to the broader understanding of the divisibility properties of partition functions with multiple constraints. The work is presented as a continuation of the authors' previous research [13], offering new infinite families and specific congruences that complement earlier findings. The paper does not propose new combinatorial interpretations or applications beyond the arithmetic properties of the sequence itself.