$t$-Regular Partition Theory

• $t$-regular partitions are defined as partitions where no part is divisible by $t$, playing a fundamental role in combinatorics and number theory.
• Generating functions, explicit bijections such as Glaisher's theorem, and Euler-type recurrences provide robust methods to analyze and count $t$-regular partitions.
• Recent studies highlight monotonicity in hook-length statistics and deep congruence phenomena, opening pathways to connections with modular forms and advanced partition theory.

A $t$-regular partition is a partition theoretic object fundamental to enumerative combinatorics, number theory, and representation theory, with intricate connections to classical partition identities and deep arithmetic properties. Formally, a $t$-regular partition of a nonnegative integer $n$ is a partition in which no part is divisible by $t$, i.e., for every part $\lambda_i$ in the partition, $\lambda_i \not\equiv 0 \pmod t$. The study of $t$-regular partitions encompasses generating function methods, explicit bijections (notably Glaisher's), hook-length statistics, monotonicity inequalities, and nontrivial congruence phenomena, forming a nexus between classical algebraic combinatorics and modern analytic number theory.

1. Definitions and Foundational Properties

A partition $$\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r > 0)$$ of $n$ is $t$-regular if $\lambda_i \not\equiv 0 \pmod{t}$ for all $i$. The set of all $t$-regular partitions of $n$ is denoted $B_t(n)$. Example: the $2$-regular partitions of $7$ consist of those with all odd parts; for $t=3$, no part may be a multiple of $3$.

The ordinary generating function for $t$-regular partitions is

$$G_t(q) = \sum_{n \ge 0} B_t(n) q^n = \prod_{m=1}^\infty \frac{1 - q^{t m}}{1 - q^m}$$

which removes the contributions of parts divisible by $t$ from the standard partition function, equivalently expressed as $\prod_{m \not\equiv 0 \pmod{t}} (1-q^m)^{-1}$ (Sriram et al., 30 Sep 2025, Lin et al., 18 Nov 2025, Singh et al., 2022, Barman et al., 2022).

A central combinatorial result is Glaisher's theorem: for $t \geq 2$, the number of $t$-regular partitions equals the number of $t$-distinct partitions (where no part repeats more than $t-1$ times), i.e. $|B_t(n)|=|A_t(n)|$. The explicit bijection proceeds by splitting multiplicities in $t$-distinct partitions via base-$t$ expansion, mapping them to $t$-regular partitions (Lin et al., 18 Nov 2025).

2. Generating Functions, Recurrences, and Analytic Identities

The generating function $G_t(q)$ admits several analytic forms:

• Product form: $\prod_{m=1}^\infty (1-q^{t m})/(1-q^m)$
• Eta-quotient: $q^{-(t-1)/24} \eta(tz)/\eta(z)$, with Dedekind eta $\eta(z)$ (Barman et al., 2022).

The $t$-regular partition function $p_t(n)$ admits Euler-type recurrences derived from pentagonal number expansions: $$p_t(n) = \sum_{k \neq 0} (-1)^{k+1} p_t(n-w_k) \quad \text{if } n \notin t\{w_j\}$$ with corrections when $n$ is a multiple of a pentagonal number by $t$ ($w_k = k(3k-1)/2$), generalizing Euler's recurrence for the unrestricted partition function (Bhowmik et al., 2024).

3. Monotonicity and Hook-Length Statistics

For $k \ge 1$, let $b_{t,k}(n)$ denote the total number of hooks of length $k$ in all $t$‐regular partitions of $n$, i.e.,

$$b_{t,k}(n) = \sum_{\lambda \vdash_t n} |\{ c \in \lambda : h(c) = k \}|$$

where $h(c)$ is the hook-length at cell $c$ in the Ferrers–Young diagram.

Recent work establishes several monotonicity ("hook-bias") inequalities in $t$:

• $b_{t+1,1}(n) \geq b_{t,1}(n)$ for all $n$
• $b_{3,2}(n) \geq b_{2,2}(n)$ for $n>3$
• $b_{3,3}(n) \geq b_{2,3}(n)$ for all $n$
• $b_{t m, k}(n) \geq b_{t, k}(n)$ for $m \ge 1$ (Singh et al., 2024)

Combinatorial proofs utilize injective maps between partition sets, preserving or increasing the relevant hook statistics. For example, the mapping $Q_{t,n}: B_t(n) \to B_{t+1}(n)$ splits parts divisible by $t+1$, increasing the number of distinct parts and so the $1$-hook total. Partition families are decomposed and matched via injections or paired to ensure counts of $k$-hooks do not decrease with increasing $t$.

Explicit generating functions for $b_{t,k}(n)$ for $k=1,2,3$ have been derived; for instance,

$$\sum_{n \ge 0} b_{t,2}(n) q^n = \frac{(q^{t};q^{t})_\infty}{(q;q)_\infty} \frac{q^{2t-1} - q^{2t} + q^{2t+1}}{(1-q^2)(1-q^t)}$$

(Lin et al., 30 Apr 2025). Explicit bounds $N = O(t^5)$ are established for thresholds beyond which $b_{t,2}(n) \geq b_{t,1}(n)$ holds universally.

4. Congruence Phenomena, Self-Similarity, and Parity Distributions

$t$-regular partition functions manifest deep congruences and self-similarity patterns modulo primes, generalizing Ramanujan-type congruences for the unrestricted partition function. For $b_t(n) = |B_t(n)|$, infinite families of congruences modulo $2$, $3$, $5$, and other primes are established (Barman et al., 2022, Singh et al., 2022):

• For $t=9$, primes $p\not\equiv \pm 1 \pmod 9$, and suitable $a$,

$$\sum_{n} b_9(2(pn+a)) q^n \equiv q^8 \sum_{n} b_9(2n+1) q^{pn} \pmod{2}$$

• For general $t$, lifts of partition congruences: if $p(an+b) \equiv 0 \pmod{m}$, then $b_{a t}(an+b) \equiv 0 \pmod{m}$ for all $n$ (Barman et al., 2022).

Parity-distribution results quantify the frequency of even and odd values of $b_t(n)$, with lower bounds on the density of even coefficients for certain $t$, and stability properties for odd densities in infinite families of eta-quotients related to $t$-regular partition generating functions.

5. Generalizations: $t$-Distinct and Refined Partition Families

The theory extends to $t$-distinct partitions $A_t(n)$ where no part occurs more than $t-1$ times. Glaisher's bijection equates $|A_t(n)|=|B_t(n)|$, and explicit generating functions are available: $$\sum_{n \ge 0} A_t(n) q^n = \prod_{m=1}^\infty \frac{1 - q^{t m}}{1 - q^m}$$ (Lin et al., 18 Nov 2025). Further refinements such as the $E_t(n)$ family combine restrictions on the largest part and multiplicities, with $|E_t(n)| = |B_t(n)|$ via explicit bijection.

Hook-length inequalities and monotonicity also extend to $t$-distinct partition statistics ($d_{t,k}(n)$), yielding $d_{t+1,k}(n) \geq d_{t,k}(n)$ (Singh et al., 2024).

6. Open Problems and Future Directions

Current research highlights several open questions:

• Generalizing hook-length monotonicity: for fixed $k$, prove $b_{t+1,k}(n) \geq b_{t,k}(n)$ for all $n$ and $t$ beyond small $k$.
• Developing explicit (analytic or $q$-series based) proofs for hook-length inequalities (e.g., establishing $b_{3,2}(n) \geq b_{2,2}(n)$ via generating functions).
• Determining sharp thresholds $N_{t,k}$ for universal inequalities in $n$.
• Closed-form expressions for generating functions of $b_{t,k}(n)$ for all $t, k$.
• Exploring further the interaction with modular forms, Hecke operators, and eta-quotient arithmetic.

These open directions motivate applications to modular representation theory, investigation of $t$-core partitions, and understanding modular forms associated with hook-length statistics, continuing the interface between algebraic combinatorics, analytic number theory, and representation theoretic structures (Singh et al., 2024, Lin et al., 30 Apr 2025).