Arithmetic Properties of Overcolored Odd Partitions

Technical Summary: Arithmetic Properties of Overcolored Odd Partitions

Problem Statement and Context
This paper investigates the arithmetic properties of a specific partition function, denoted $\bar{a}_s(n)$. This function counts the number of partitions of a positive integer $n$ where each odd part is assigned one of $s$ distinct colors (where $s \ge 1$), and the first occurrence of any part may be overlined. This object serves as a natural extension of the classical overpartition function $\bar{p}(n)$, introduced by Corteel and Lovejoy, and the colored partition function $p_2(n)$, where odd parts appear in two colors. The authors note that $p_2(n)$ is equivalent to $\bar{p}(n)$, and $\bar{a}_s(n)$ generalizes these concepts by allowing $s$ colors for odd parts while retaining the overlining feature.

The primary objective is to establish new infinite families of congruences for $\bar{a}_s(n)$ modulo powers of 2. Specifically, the paper seeks to identify conditions under which $\bar{a}_s(n)$ vanishes modulo $2^k$ for various $k$, depending on the parameters $s$, the prime $p$, and the structure of $n$.

Methodology
The authors employ a combination of generating function manipulations, the theory of modular forms, and specific results by Newman regarding eta-quotients.

1. Generating Functions: The generating function for $\bar{a}_s(n)$ is derived as:
   $$\sum_{n=0}^{\infty} \bar{a}_s(n)q^n = \frac{f_2^{3s-2}}{f_1^{2s} f_4^{s-1}}$$
   where $f_m = (q^m; q^m)_\infty$. The proofs rely heavily on dissecting this generating function into components based on the residue of the exponent modulo powers of 2 (e.g., $2n+1$, $4n+1$, $8n+5$).

2. Modular Forms and Eta-Quotients: The paper utilizes the theory of modular forms on $\Gamma_0(N)$. The generating functions are expressed as eta-quotients. The authors apply theorems by Gordon-Hughes and Newman to confirm that these quotients are holomorphic modular forms. They further utilize the properties of Hecke operators and Hecke eigenforms to analyze the coefficients of these series.

3. Dissection and Congruence Lemmas: A core technical component involves proving specific dissection formulas (Lemmas 3.1–3.4) that express the generating functions of $\bar{a}_s(n)$ restricted to arithmetic progressions (e.g., $4n+1$, $8n+7$) in terms of simpler eta-quotients modulo powers of 2. These lemmas establish congruences such as $\sum \bar{a}_{2k\alpha}(2n+1)q^n \equiv 2^{k+1}f_1^{12} \pmod{2^{k+2}}$.

4. Newman's Results: The proofs of the main theorems rely on recursive identities for coefficients of eta-quotients derived from Newman's work (Lemmas 2.5–2.8). These identities relate coefficients at indices $np + \delta$ to coefficients at $n$ and $n/p$, allowing the authors to propagate congruences through powers of a prime $p$.

Key Contributions and Results
The paper establishes several theorems describing infinite families of congruences for $\bar{a}_s(n)$ modulo powers of 2. The results are categorized by the parity of the color parameter $s$ (specifically $s=2m\alpha$ vs. $s=2\beta+1$) and the properties of the prime $p$.

• Theorems 1.1 and 1.2: Establish congruences modulo $2^{m+2}$ and $4$ for $\bar{a}_{2m\alpha}(n)$ and $\bar{a}_{2\beta+1}(n)$ respectively. These hold for primes $p \ge 3$ (with specific conditions like $p \not\equiv 1 \pmod 4$) and rely on the initial condition that $\bar{a}_s(p) \equiv 0 \pmod 2$. The congruences apply to arguments of the form $2p^{2k+1}n + p^{2k+1}$ and products of squares of primes.
• Theorem 1.4 and Corollary 1.5: Provide recursive relations for primes $p \equiv 3 \pmod 4$, linking $\bar{a}_s(4p^{k+1} + \dots)$ to $\bar{a}_s(4p^{k-1} + \dots)$. This leads to a general formula involving powers of $p^2$.
• Theorems 1.6–1.9: Extend these results to different arithmetic progressions (e.g., $4p^{2k+1}n + p^{2k+2}$, $8p^{2k+1}n + 4p^{2k+1}$) for various primes $p \ge 3$ and $p \ge 5$, yielding congruences modulo $2^{m+2}$, $4$, $8$, and $16$.
• Theorems 1.10 and 1.12: Introduce more complex conditions involving a parameter $\kappa(p)$ defined via coefficients of specific eta-quotients ($f_1^3 f_2^6$ and $f_1 f_2^{10}$). These theorems establish congruences modulo $2^{m+3}$, $16$, $2^{m+4}$, and $128$ based on the parity of $\kappa(p)$.
• Theorem 1.11: Addresses the case where $p \equiv 1 \pmod 4$ and $s$ is a general positive integer, establishing a congruence modulo 16.

Significance and Claims
The authors state that the paper's primary contribution is the establishment of "new infinite families of congruences" for the partition function $\bar{a}_s(n)$. The work builds upon existing literature regarding overpartitions and colored partitions, specifically extending the connections explored by Andrews and Mohamed El Bachraoui.

The significance lies in the systematic application of modular form theory (specifically Hecke eigenforms and eta-quotients) to derive these congruences. The paper does not claim to solve a broader open problem in partition theory but rather adds specific, rigorous results to the growing body of literature on the arithmetic properties of overpartitions and their generalizations. The results are presented as direct consequences of the generating function manipulations and the application of Newman's theorems, without positing new applications or experimental proposals.