Arithmetic Properties for $k$-Color Analogue of Simultaneously $s$-Regular and $t$-Distinct Partitions

Technical Summary: Arithmetic Properties for k-Color Analogue of Simultaneously s-Regular and t-Distinct Partitions

Problem Statement
This paper investigates the arithmetic properties of a specific class of integer partitions: the $k$-colored analogue of partitions that are simultaneously $s$-regular and $t$-distinct. While the uncolored partition function $RD_{\ell,t}(n)$, introduced by William Keith, counts partitions of $n$ where no part is divisible by $\ell$ and no part appears $t$ or more times, this study extends the concept to $k$-colored partitions. In this context, a partition is $k$-colored, $s$-regular, and $t$-distinct if parts are not divisible by $s$, appear fewer than $t$ times, and each part can appear in one of $k$ distinct colors. The authors focus specifically on the case where the number of colors is $k=3$, examining the generating functions and congruence properties of $RD_{3,3}^3(n)$ and $RD_{3,27}^3(n)$.

Methodology
The primary methodology employed is the $H$-operator approach, specifically utilizing the $H_3$ operator (a dissection operator that extracts terms where the exponent of $q$ is a multiple of 3), following the techniques established by Michael D. Hirschhorn.

The technical framework relies on several key components:

1. Generating Functions: The generating function for $RD_{\ell,t}^k(n)$ is established as $\left( \frac{E_\ell E_t}{E_1 E_{\ell t}} \right)^k$, where $E_r = \prod_{n=1}^\infty (1-q^{nr})$.
2. Modular Equations: The authors utilize Jacobi's identity and derive a modular equation for the variable $\zeta = E_1^3 / (q E_9)$. This leads to the cubic relation $\zeta^3 + 9\zeta^2 + 27\zeta - T = 0$, where $T = E_3^{12} / (q^3 E_9^{12})$.
3. Recursive Matrix Structures: By applying the $H_3$ operator to powers of $1/\zeta$, the authors derive a recursive relationship involving an infinite matrix $M = (m_{i,j})$. This matrix governs the coefficients of the resulting series expansions.
4. Inductive Proofs: The core of the proof involves establishing recurrence relations for coefficient vectors ($x_k$ and $y_k$) using matrix multiplication ($x_{k+1} = x_k \cdot A$). These relations allow the authors to express the generating functions for specific arithmetic progressions of $n$ in terms of $q$-series involving powers of $E_3$ and $E_1$.
5. Valuation Analysis: To prove congruences, the authors analyze the $3$-adic valuation ($\nu$) of the matrix entries and the resulting coefficient vectors, demonstrating that the coefficients are divisible by increasing powers of 3.

Key Contributions and Results
The paper establishes infinite families of congruences modulo powers of 3 for the 3-colored partition functions. The main results are:

• Theorem 1.1 (Case $\ell=3, t=3$): The authors derive a general formula for the generating function of $RD_{3,3}^3$ evaluated at arguments of the form $3^k n + \frac{3^k+1}{2}$. The result is expressed as a finite sum involving coefficients $x_{k,i}$ derived from a matrix recurrence.
  • Corollary 1.2: From this theorem, it is deduced that $RD_{3,3}^3\left(3^k n + \frac{3^k+1}{2}\right) \equiv 0 \pmod{3^{k+1}}$. This is presented as an analogue to Ramanujan's famous congruences for the standard partition function $p(n)$ modulo powers of 5.
• Theorem 1.3 & 1.5 (Case $\ell=3, t=27$): The study extends to the case where $t=27$. The authors provide explicit generating functions for $RD_{3,27}^3(3n+2)$ and $RD_{3,27}^3(9n+2)$, and subsequently generalize this to an infinite family for arguments of the form $3^k n + \frac{3^k+13}{2}$ for $k \ge 3$.
  • Corollary 1.6: This leads to the congruence $RD_{3,27}^3\left(3^k n + \frac{3^k+13}{2}\right) \equiv 0 \pmod{3^{k+1}}$ for $k \ge 3$.

Specific examples are provided to illustrate the theory, such as the identity for $k=1$ and $k=2$ in Theorem 1.1, which are noted to be structurally analogous to Ramanujan's identities for $p(5n+4)$ and $p(7n+5)$.

Significance and Claims
The paper claims to initiate the systematic study of the $k$-colored analogue of the $RD_{\ell,t}$ function. Its significance lies in:

1. Extension of Known Theory: It generalizes previous results on $RD_{\ell,t}$ (specifically those by Nadji and Ahmia) to the multi-colored setting.
2. New Congruence Families: It provides new, infinite families of congruences modulo powers of 3, which are rare and difficult to obtain in partition theory compared to congruences modulo 5 or 7.
3. Structural Analogies: The authors highlight that their derived identities and congruences bear a strong structural resemblance to Ramanujan's most celebrated results, suggesting a deep underlying modular structure in these colored partition functions.

The work is strictly theoretical, focusing on the derivation of generating functions and the proof of arithmetic properties via $q$-series manipulations and operator methods, without proposing experimental applications or future extensions beyond the scope of the proven theorems.