Families of Congruences for Partitions with $k$-colored odd parts

1. Problem Statement

The paper investigates the arithmetic properties of a specific class of integer partitions denoted as $a_k(n)$. These are partitions of an integer $n$ where:

• Odd parts may appear in $k$ distinct colors.
• Even parts are restricted to at most one color (uncolored).

The generating function for $a_k(n)$ is given by:
$$\sum_{n=0}^{\infty} a_k(n)q^n = \frac{f_2^{k-1}}{f_1^k} = \prod_{n=1}^{\infty} \frac{(1-q^{2n})^{k-1}}{(1-q^n)^k}$$
where $f_k = \prod_{n=1}^{\infty} (1-q^{kn})$.

While previous work by Hirschhorn, Sellers, Thejitha, and Fathima had established specific congruences for these functions (particularly modulo 3), the goal of this paper is to generalize these results to identify infinite families of congruences modulo 3 for a broader set of parameters, and to explore Ramanujan-type congruences for specific cases like $k=11$.

2. Methodology

The author employs a combination of modular form theory, Hecke operators, and computational verification to prove the main results. The methodology follows a structured approach:

• Modular Forms and Eta-Quotients: The generating functions are manipulated into eta-quotients. The paper utilizes the criteria (Lemma 2.1) to determine when an eta-quotient constitutes a modular form in the space $M_k(\Gamma_0(N), \chi)$.
• Hecke Operators: The core technique involves using Hecke operators ($T_p$) to "lift" base congruences to higher powers. Specifically, the author uses the property that modulo a prime $p$, the action of $T_p$ on a Fourier series $f(z) = \sum a(n)q^n$ simplifies to $\sum a(pn)q^n$.
• Inductive Structure: The proof strategy relies on establishing two types of congruences:
  1. Base Congruence: A congruence holding for a specific initial modulus (e.g., $27n + C$).
  2. Internal Congruence: A relation that allows the argument of the partition function to be scaled (e.g., relating $a(9n+t)$ to $a(81n + \dots)$).
     By combining these via induction, infinite families of congruences are generated.
• Sturm Bound: To prove that two modular forms are congruent modulo 3, the author verifies that their Fourier coefficients match up to the Sturm bound (Lemma 2.2). This reduces an infinite verification problem to a finite computational check, which was performed using Maple.

3. Key Contributions and Results

A. Generalization of Infinite Families (Theorem 1.3)

The primary contribution is Theorem 1.3, which generalizes previous results by Sellers and Thejitha/Fathima.

• The Result: For a specific set of integers $\alpha \in \{1, 3, 4, \dots, 63\}$, the author proves an infinite family of congruences modulo 3.
• The Formula: For any $\alpha = 9j + t$ (where $0 \le t \le 8$) and for all $n, k \ge 0$:
  $$a_{3\alpha+2}\left(3^{2k+3}n + 9^k(18+t) + \frac{\alpha 9^k - 1}{8}\right) \equiv 0 \pmod 3$$
• Mechanism: The proof relies on Lemma 3.1 (inductive step) and Lemma 3.2 (construction of modular forms $g_1$ and $g_2$). Lemma 3.2 establishes that specific eta-quotients involving $E_4(z)$ and Hecke operators $T_3^2$ and $T_3^4$ yield the necessary internal congruences.

B. Ramanujan Congruences for $a_{11}(n)$ (Theorem 4.2)

The paper derives specific Ramanujan-style congruences for the case $k=11$:

• $a_{11}(5n + 4) \equiv 0 \pmod 5$
• $a_{11}(7n + 4) \equiv 0 \pmod 7$
• $a_{11}(11n + 1) \equiv 0 \pmod{11}$
  The proofs utilize dissection techniques and the known Ramanujan congruences for the standard partition function $p(n)$, combined with the specific generating function structure of $a_{11}(n)$.

C. Computational Verification

A significant portion of the work involves verifying the "orders of vanishing" at cusps for the constructed modular forms and checking the congruence of Fourier coefficients up to the Sturm bound using Maple. This confirms that the theoretical conditions for the infinite families hold for the specific $\alpha$ values listed.

4. Significance and Future Directions

• Unification: The paper unifies disparate results in the literature regarding $k$-colored partitions with restricted even parts, providing a unified framework for modulo 3 congruences.
• Methodological Advancement: It demonstrates the efficacy of using Hecke operators on eta-quotients to generate infinite families of congruences, a technique that can be applied to other partition functions.
• Open Questions: The author concludes with three critical questions for future research:
  1. Why are certain values of $\alpha < 81$ missing from the infinite family list?
  2. Can the internal congruences (which seem to stop around $\alpha \approx 70$) be extended infinitely for other $\alpha$?
  3. Do the Ramanujan congruences found for $a_{11}$ belong to a broader infinite family for other $k$?

In summary, this paper advances the understanding of the arithmetic structure of colored partitions by rigorously proving new infinite families of congruences modulo 3 and establishing classical Ramanujan congruences for specific cases, utilizing the powerful machinery of modular forms.