Lin’s Restricted Partition Function

Updated 16 October 2025

Lin’s restricted partition function is a family of partition-counting functions defined by constraints on the parts, represented by explicit q-series generating functions.
The analytic methods employ infinite product representations and modular form transformations to derive explicit formulas, combinatorial interpretations, and arithmetic congruences.
Recent advances reveal log-concavity, quasi-polynomial structures, and Fibonacci-like recurrences, offering new insights into additive number theory and partition statistics.

Lin’s restricted partition function refers to various families of partition-counting functions in which specific combinatorial or arithmetic restrictions are enforced on the underlying parts or structure of the partition. These functions have motivated diverse research themes, including modular forms, $q$ -series, quasi-polynomial structure, log-concavity, and congruence properties. Recent literature has advanced explicit formulas, combinatorial interpretations, and deep arithmetic congruence phenomena for these functions.

1. Definitions and Prototypical Constructions

The term “Lin’s restricted partition function” encompasses several models, notably:

Three-colored restricted partitions: $b(n)$ is the number of triples $\pi = (\pi_1, \pi_2, \pi_3)$ , with $\pi_1$ a partition into distinct odd parts, and $\pi_2, \pi_3$ partitions whose parts are all divisible by $4$ (Guadalupe, 31 Mar 2025).
Color and parity modifications: Variants such as $B(n)$ , where two colors require distinct odd parts and the third color requires parts divisible by $4$ (Guadalupe, 15 Oct 2025).
Restricted $m$ -ary, multiplicity- or alphabet-constrained, and bounded largest part partitions: For example, $p_{\mathcal{A}}(n, k)$ , counting partitions into multiset $\{a_1,\ldots,a_k\}$ , or $p_N(n)$ , the number of partitions of $n$ into at most $N$ parts (Gajdzica, 2021, Kiran, 2022, Gajdzica, 2023).

The underlying structure is encoded in the generating functions, which typically have the form:

$G(q) = \prod_{i} \frac{f_{\text{numerator}}(q)}{f_{\text{denominator}}(q)},$

where $f_{m} := \prod_{n\geq1} (1 - q^{mn})$ , and the exponents and indices are dictated by the partition restrictions.

2. Generating Functions and Analytic Approaches

The generating functions for these restricted partition functions admit representations in terms of infinite products, $q$ -series, and occasionally theta functions. For example:

For Lin’s $b(n)$ function:

$\sum_{n=0}^\infty b(n) q^n = \frac{f_2}{f_1f_4},$

where $f_k = \prod_{n\geq1}(1 - q^{kn})$ . This encodes partitions with one component of distinct odd parts ( $f_2/f_1$ ), and two components with parts divisible by $4$ ( $1/f_4$ ) (Guadalupe, 31 Mar 2025).

For the analogue $B(n)$ ,

$\sum_{n=0}^\infty B(n) q^n = \frac{f_2^4}{f_1^2 f_4^3} [2510.13685].$

The manipulation and dissection of these generating functions via classical $q$ -series identities (e.g., Euler’s, 2- or 3-dissection, Hecke operator theory, eta-quotient modular transformations) are central for extracting explicit enumeration and congruence relations.

3. Explicit Formulas, Quasi-Polynomial Structure, and Polynomial Part

Restricted partition functions for a finite set or with bounded largest part generally manifest as quasi-polynomials in $n$ , with period depending on the arithmetic structure of the restriction (often the least common multiple of the part sizes):

$p_{\mathbf{a}}(n) = d_{\mathbf{a},r-1}(n) n^{r-1} + d_{\mathbf{a},r-2}(n) n^{r-2} + \cdots + d_{\mathbf{a},0}(n),$

with $d_{\mathbf{a},m}(n)$ periodic and computable either from closed sums over solutions to relevant congruences or from explicit determinant or Cramer’s rule formulas involving Bernoulli polynomials and Barnes–Bernoulli numbers (Cimpoeas et al., 2016, Cimpoeas, 2018, Cimpoeas, 2019).

The generating function approach leads, for example, to explicit Sylvester's wave decompositions, where

$p_N(n) = \sum_{k=1}^N W_k(n; N),$

with each $W_k$ a $k$ -periodic quasi-polynomial. Recent work connects the coefficients of the $q$ -partial fractions for these waves to linear combinations of Ramanujan sums, degenerate Bernoulli and Euler numbers, and introduces new analytic invariants such as the Gaussian–Ramanujan sum (Kiran, 2022).

The polynomial part of such quasi-polynomials can be isolated via averaging over a period or via auxiliary analytic techniques.

4. Modular and Arithmetic Congruence Phenomena

A major line of investigation is the derivation of Ramanujan-type congruences for restricted partition functions, akin to $p(5n+4)\equiv0\pmod{5}$ . For $b(n)$ , $B(n)$ , and related functions, one finds:

$b(3n+2)\equiv0\pmod{3}$ , via the structure of the generating function (Guadalupe, 31 Mar 2025).
$B(27n+16)\equiv0\pmod{3}$ , $B(5n+4)\equiv0\pmod{5}$ , $B(2n+1)\equiv0\pmod{2}$ , and further congruences involving finite weighted sums of shifted values (Guadalupe, 15 Oct 2025).

The analytic proof techniques combine $q$ -series dissections (extracting progressions by expanding products in different moduli) and modular form transformations. For certain cases, the use of Radu’s Ramanujan-Kolberg algorithm, as implemented in computer algebra packages, establishes congruences for eta-quotient series by leveraging the modular curve structure.

Additionally, infinite families of “internal” congruences relate widely separated values, further reflecting deeper arithmetic symmetry, e.g., $b(729n+456)\equiv b(9n+6)\pmod{3}$ (Guadalupe, 31 Mar 2025).

5. Structural Properties: Log-Concavity, Multiplicativity, and Fibonacci-like Recurrences

Log-concavity and strong log-concavity (and more generally, $r$ -log-concavity) have been established for many restricted partition functions, including those with a finite alphabet or restricted largest part, under arithmetic coprimality hypotheses (Gajdzica, 2023, Roy, 4 Apr 2024). This property yields multiplicative abundance: for sufficiently large $n, m$ ,

$f(n)f(m)\geq f(n+m)$

with strict inequality except in finitely many cases (Roy, 4 Apr 2024). These findings unify and generalize classical results of DeSalvo–Pak, Bessenrodt–Ono, and others for unrestricted partitions.

In certain highly-structured restriction schemes, exact Fibonacci-type equalities replace inequalities, i.e.,

$p(n|\text{parts}\equiv 12,15,27 \bmod 27) = p(n-1|\text{parts}\equiv 6,21,27) + p(n-2|\text{parts}\equiv 3,24,27)$

and generalizations to higher truncations of Euler’s recurrence exist (Zheng, 2023).

6. Combinatorial and Analytical Implications

The paper of Lin’s restricted partition functions, their analogues, and their arithmetic properties yields several significant consequences:

Combinatorial enumeration: The explicit product and sum formulas enable highly granular enumeration results for partition functions under fine restrictions.
Arithmetic symmetries: The congruence results parallel and extend Ramanujan’s congruences, often involving elaborate progressions arising from the structure of the generating function.
Algorithmic enumeration and density: For functions with finite support (e.g., bounded largest part), periodicity and explicit recurrences enable density and modular residue analyses (Gajdzica, 2021).
Connections to modular forms: The generating functions for these combinatorial invariants are often explicit eta-quotients or modular forms on congruence subgroups, making them accessible to the deep machinery of modular and automorphic function theory.
Generalizations in additive number theory: Techniques developed for $p_{\mathbf{a}}(n)$ and counterparts play crucial roles in the paper of the generalized Frobenius problem, representation asymptotics, and level set tilings in numerical semigroups (Woods, 2020).

7. Principal Methods and Future Directions

Methodologically, current research exploits:

$q$ -series dissections and identities,
modular forms and eta-quotient transformations,
determinant evaluations involving Bernoulli and Barnes–Bernoulli numbers,
partial fraction decompositions (Sylvester waves),
analytic and algebraic tools for quasi-polynomials,
stochastic-combinatorial constructions for uniqueness or sparse enumeration (Alon, 2012).

A plausible implication is the extension of these techniques to increasingly complex partition statistics: higher colorings, additional restrictions (e.g., designated summands, overlining, parity/length, etc.), and connection with deep arithmetic (e.g., $p$ -adic modular forms, higher degree congruences).

Open problems include deeper combinatorial proofs of congruences (in particular, bijective or sign-reversing involution arguments), extension of existing congruences to higher moduli or other primes, and the full characterization of log-concave or multiplicatively abundant restricted partition sequences for arbitrary combinatorial constraints.

Table 1: Generating Functions for Prototypical Lin's Restricted Partition Functions

Function	Generating Function	Notable Restrictions
$b(n)$	$\displaystyle \sum_{n=0}^\infty b(n)q^n = \frac{f_2}{f_1f_4}$	Distinct odd parts; parts divisible by 4
$B(n)$	$\displaystyle \sum_{n=0}^\infty B(n)q^n = \frac{f_2^4}{f_1^2f_4^3}$	Two sets of distinct odd parts; parts $\equiv0\!\pmod{4}$
General $p_{\mathcal{A}}(n,k)$	$\displaystyle \prod_{i=1}^k \frac{1}{1-q^{a_i}}$	Partition into multiset $\{a_1,\dots,a_k\}$
$p_N(n)$	$\displaystyle \prod_{i=1}^N \frac{1}{1-q^i}$	At most $N$ parts

The rigorous paper of Lin’s restricted partition function and its analogues thus weaves together algebraic, combinatorial, and arithmetic methodologies, illustrating the breadth and depth of modern partition theory.