A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi--Colored Odd Parts

Technical Summary: A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi-Colored Odd Parts

Problem Statement
The paper addresses the arithmetic properties of a specific family of integer partition functions, denoted $a_k(n)$. These functions count the number of partitions of an integer $n$ where even parts are monochromatic (appearing in only one color) and odd parts are "colored" with one of $k$ available colors. This generalizes the work of Amdeberhan and Merca, who studied the specific case $a_3(n)$ (where odd parts have three colors) and proved the congruence $a_3(7n+2) \equiv 0 \pmod 7$. While Amdeberhan and Merca established this result using the computer algebra package RaduRK, the authors of this paper seek to provide a purely elementary proof and extend these findings to an infinite family of related functions.

Methodology
The authors employ elementary generating function manipulations and classical $q$-series identities. The core of their methodology relies on analyzing the generating functions for $a_k(n)$ modulo 7.

1. Generating Function Formulation: The generating function for the family is established as:
   $$\sum_{n=0}^{\infty} a_k(n)q^n = \frac{f_2^{k-1}}{f_1^k}$$
   where $f_k := (q^k; q^k)_\infty$ is the standard $q$-Pochhammer symbol shorthand.

2. Modular Reduction Tools: The proofs utilize specific lemmas derived from classical results (primarily from the first author's previous work [10] and Jacobi's identities):

   • Lemma 2.1 (Jacobi): Provides a congruence for $f_1^3$ modulo 7, showing it reduces to a sum of terms involving powers of $q^7, q^8,$ and $q^{10}$ (specifically $J_0(q^7) + qJ_1(q^7) + q^3J_3(q^7)$).
   • Lemma 2.3: Establishes congruences for ratios of $f$ terms (such as $f_1^5/f_2^2$ and $f_1^2 f_4^2/f_2$) modulo 7, demonstrating that they reduce to series containing only powers of $q$ congruent to $0, 1,$ or $5 \pmod 7$.
   • Lemma 2.4: Utilizes the binomial theorem property $f_a^{bp} \equiv f_{ap}^b \pmod p$ for a prime $p$.

3. Proof Strategy: For each specific case in the main theorem, the authors substitute the generating function into the modular framework. By replacing $q$ with $q^2$ or $q^7$ where necessary and applying the lemmas, they demonstrate that the resulting series expansion lacks terms with exponents congruent to specific residues modulo 7.

Key Contributions and Results
The primary contribution is the proof of Theorem 1.3, which establishes infinitely many congruences modulo 7 for the family $a_k(n)$. Specifically, for all $n \ge 0$:

• $a_1(7n + 5) \equiv 0 \pmod 7$ (a restatement of Ramanujan's congruence for $p(n)$).
• $a_3(7n + 2) \equiv 0 \pmod 7$ (re-proving the Amdeberhan-Merca result elementarily).
• $a_4(7n + 4) \equiv 0 \pmod 7$.
• $a_5(7n + 6) \equiv 0 \pmod 7$.
• $a_7(7n + 3) \equiv 0 \pmod 7$.

Furthermore, the authors generalize these findings in Corollary 4.1, showing that for any integer $j \ge 0$, the congruences hold for the sub-families $a_{7j+k}(n)$ with the same residue conditions on $n$. This demonstrates that the divisibility by 7 persists even as the number of colors on the odd parts increases without bound.

Significance and Claims
The paper claims significance primarily in the realm of proof methodology and the extension of known results:

• Elementary Proof: The authors explicitly state that a primary goal is to provide a "truly elementary proof" of the Amdeberhan-Merca congruence ($a_3(7n+2) \equiv 0 \pmod 7$), avoiding the reliance on computer algebra systems like RaduRK.
• Unification: The work places the specific function $a(n)$ within a broader, infinite family of partition functions, revealing a unified structure of congruences modulo 7.
• Persistence of Divisibility: The authors highlight that it is not immediately obvious from the combinatorial definition that such strong divisibility properties would persist as the number of colors ($k$) increases. The paper demonstrates that these congruences are robust across the infinite family.

The paper concludes by noting that while the extension to an infinite family (Corollary 4.1) is proved with relative ease, the existence of these congruences for arbitrarily large $k$ is a non-trivial arithmetic property of these colored partitions. The authors also briefly mention the related problem of swapping the roles of even and odd parts (monochromatic odd parts, multi-colored even parts), citing recent work by other researchers in that specific direction.