Infinite Families of Ramanujan-like Congruences

• Infinite families of Ramanujan-like congruences are systematic arithmetic patterns where q-series coefficients vanish in specific modular progressions, showcasing deep partition symmetries.
• They employ advanced techniques such as q-series dissections, modular forms, and operator theory to derive precise congruence formulas for various partition functions including k-regular partitions and overpartitions.
• These congruences extend Ramanujan’s classical results, linking combinatorial partition identities with arithmetic geometry and opening avenues for algorithmic verification and new theoretical insights.

Infinite families of Ramanujan-like congruences are systematic, arithmetic vanishing patterns satisfied by the coefficients of a broad class of partition-theoretic and q-series generating functions, extending the phenomenon first observed by Ramanujan for the ordinary partition function. This area encompasses congruences for a diverse range of partition functions—including k-regular partitions, generalized Frobenius partitions, colored compositions, partition tuples with t-cores, overpartitions, and specialized generating functions arising from modular and theta identities—with the structure, mechanism, and generality of these congruence families deeply informed by modular forms, q-dissections, operator theory, and arithmetic geometry.

1. Definition and Structural Scope

The term “infinite family of Ramanujan-like congruences” refers to an infinite collection of congruences for partition functions (or their relatives), typically of the form

$$f(Mn + r) \equiv 0 \pmod{m} \qquad \text{for all } n \ge 0,$$

where $f(n)$ is the arithmetic function of interest, $M$ the modulus of progression (often a prime power or product of primes), and $m$ the modulus of the congruence (often, but not exclusively, a prime or its power). The prototypical examples are Ramanujan’s congruences for the partition function: $$p(5n + 4) \equiv 0 \pmod{5},\qquad p(7n + 5) \equiv 0 \pmod{7},\qquad p(11n + 6) \equiv 0 \pmod{11},$$ which have inspired considerable generalization and abstraction.

Key features of such families include:

• Parameterization in terms of progression moduli scaling with powers of a base (e.g., $5^k$, $3^{2l+1}$).
• Increasingly intricate arithmetic progressions, sometimes with explicit formulas for the residue classes, and often with combinatorial or modular underpinnings.
• Extension to functions beyond $p(n)$: $k$-regular partitions $b_k(n)$, generalized Frobenius partitions $c\phi_k(n)$, colored partitions $p_{-t}(n)$, overpartitions, MacMahon-type divisor sums, partition tuples with core restrictions ($A_{p,k}(n)$), and explicit q-products associated to Rogers–Ramanujan-type identities.

2. Main Types and Examples

Infinite congruence families have been established for a wide array of functions:

Function Family	Prototype Infinite Family	Modulus
Broken $2$-diamond partitions	$\Delta_2(243n+142)\equiv 0 \pmod{3}$	$3$
k-regular partitions	$b_{11}(n) \equiv 0 \pmod{11}$, $b_{13}(n) \equiv 0 \pmod{13}$	$11$, $13$
Generalized Frobenius partitions	$c\phi_{p^N+k}(pn + r) \equiv 0 \pmod{p}$	$p$
Colored partition functions	$p_{-t}(\ell n + a) \equiv 0 \pmod{\ell}$, for $t = \ell s - r$	$\ell$
MacMahon-type sums/divisor sums	$M_0(a, t; 3N + 2) \equiv 0 \pmod{2}$, etc.	$2,3,5,\ldots$
Overpartitions/tuples with cores	$$A_{5,4}(5^{a+1}n + 5^{a+1}-4) \equiv 0 \pmod{5^{a+4}}$$	$5^{a+4}$
Cubic/colored/correlated partitions	$A(9n+5)\equiv 0 \pmod{3}$, $A(27n+26)\equiv 0 \pmod{3}$	$3$

Further explicit families include results such as $b_4(507n+34)\equiv 0\pmod{3}$, $PD_2(2^a(4n+3))\equiv 0\pmod{4}$ for partitions with designated summands, and congruence families for coefficients of q-series derived from Rogers–Ramanujan-type identities $g_2(n), h(n), t(n), m(n), r(n), s(n)$ modulo powers of $2$ (Biswas et al., 14 Jul 2025).

3. Mathematical Mechanisms and Methodology

The derivation and explanation of these infinite families depend on several advanced mathematical tools:

• q-Series Dissections: Progression-specific congruences are often proved by explicit $k$-dissection of generating functions, isolating arithmetic progressions with vanishing coefficients (e.g., 3-dissection of triangular number generating functions for $\Delta_2(n)$).
• Modular Forms and Eta-quotients: Many of these partition functions (or suitable modifications) yield modular (or nearly modular) forms. After “twisting” the generating function by combining it with appropriate eta-quotients or theta blocks, one may invoke Hecke operator theory and modular congruences (i.e., $U$, $V$, $T$ operators) to deduce arithmetic vanishing in progressions (Jin et al., 2015, Zheng, 2023, Rolen et al., 2020).
• Vanishing and Internal Relations: Properties such as the “vanishing property” (e.g., $a(p n + r)=0$ for non-divisible $n$) provide recurrences that, when iterated, generate infinite families (Jin et al., 2015, Smoot, 2020, Abinash et al., 2019).
• Theory of Non-ordinary Primes: In the context of eta-products and modular forms, non-ordinary primes control which progressions yield congruence relations (Bevilacqua et al., 2019).
• Combinatorial and Inductive Methods: Sometimes, especially for congruences modulo small primes or powers, combinatorial decompositions or explicit induction on modular identities (e.g., via $U$-operators, as in Rogers–Ramanujan subpartitions) are sufficient (Smoot, 2020, Saikia et al., 2023).
• Partition-theoretic Interpretations: Many generating functions can be reinterpreted to count overpartitions, bipartitions, or colored partitions, which allows for congruence arguments based on direct combinatorial analysis once the modular or q-series transformation is executed.

4. Key Results Across Related Families

Several themes recur across the literature:

• Shared Progression Residues: Overlap in residue classes for congruences derived from fundamentally different objects (e.g., $\Delta_2(27n+16)\equiv 0\pmod{3}$ in both Radu–Sellers and Li–Yao–Zhai's families) indicates deeper arithmetic structure.
• Lifting and Amplification: Given a base congruence family (say, for $p$-cores), one can often “lift” to congruences for higher values of $k$ or stronger modulus using iterative operator identities or by formal recurrences ((Saikia et al., 2023), Theorems 1.1 and 1.6).
• Relation to Classical Cases: Many modern congruence families reduce to or contain as base cases the original Ramanujan congruences, with the modular forms or combinatorial framework extended to greater generality.
• Explicit Parametrization: Some families (e.g., for $b_4(n)$ modulo $3$) give explicit progression formulas in terms of primes satisfying certain congruence conditions, elucidating both the density and computability of such congruences (Zheng, 2023).

5. Arithmetic and Combinatorial Significance

The existence of these infinite families has several profound implications:

• Partition Asymmetries: Explicit congruences document intricate periodicities and cancellations in partition counts, often revealing underlying symmetries of the generating functions’ modular or q-hypergeometric structure.
• Modularity and Galois Representations: Many families are established by analyzing modular forms modulo primes, connecting partition congruences with Galois representations (notably in cases linked to eta-quotients and Hecke eigenforms).
• Extension of Ramanujan's Philosophy: These results systematically extend Ramanujan’s initial discoveries, demonstrating they are not idiosyncratic but prototypical of a broader arithmetic landscape inhabited by a wide variety of partition-theoretic objects.

6. Open Questions and Future Directions

Recent advances raise further research questions and potential directions:

• Systematic Classification: Which partition-theoretic families admit “uniform” infinite progression congruences for general moduli, and can a full modular forms framework predict all such families?
• Crank and Rank Statistics: Although the combinatorial “crank” explains classical congruences for small primes, asymptotic analysis shows its limitations for other moduli, motivating search for new statistics or automorphic invariants (Rolon, 2013, Rolen et al., 2020).
• Non-prime Moduli and Lacunarity: Beyond primes, the theory of non-ordinary primes and the behavior of modular and mock-modular forms for composite moduli, especially powers of two or products of small primes, remain to be fully classified (Bevilacqua et al., 2019, Nyirenda et al., 2022).
• Analogues for Non-symmetric and Weighted Partitions: The extension to functions with parity, coloring, or initial repetition constraints broaden the scope but require new combinatorial and modular techniques (see partitions with initial repetitions, overpartitions, designated summands, and colored identities).
• Algorithmic Enumeration and Verification: With increasing computational power and symbolic algebra systems (e.g., Radu’s algorithm), automatic discovery, proof, and cataloguing of congruence families for arbitrary partition-theoretic generating functions becomes feasible.

7. Representative Formulas and Mechanisms

Explicit formulas typify the structure of these congruence families:

• Broken 2-diamond partitions:

$$\Delta_2(243n+142) \equiv \Delta_2(243n+223) \equiv 0 \pmod{3}$$

• Regular partitions ($\ell$-regular):

$b_{11}(p^2 n + C) \equiv 0 \pmod{11}$

with $C$ derived from parameters and underlying quadratic forms (Jin et al., 2015).

• Generalized Frobenius:

$c\phi_{p^N+k}(pn + r) \equiv 0 \pmod{p}$

• Colored partitions:

$p_{-t}(\ell n + a) \equiv 0 \pmod{\ell}$

for $t = \ell s - r$, and $a$ determined by $24a \equiv -t \pmod{\ell}$ (Locus et al., 2016).

• Overpartitions:

$B_3(15n + 7) \equiv 0 \pmod{5}$

where $B_3(n)$ is the number of almost 3-regular overpartitions (Sellers et al., 18 Nov 2024).

These formulas are proven using operator identities, modular transformations, and careful dissection of generating functions.

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In sum, infinite families of Ramanujan-like congruences encapsulate the recursive, modular, and combinatorial symmetries inherent in a wide array of partition-theoretic and q-series functions, with implications spanning the theory of modular forms, combinatorial enumeration, and arithmetic geometry. Ongoing research seeks to unify these families under a modular framework and to identify new statistics and generating functions exhibiting analogous behavior.