Hook-Lengths in t-Regular Partitions

• Hook-lengths in t-regular partitions are defined by counting hooks in partitions where no part is divisible by t, thus capturing both arithmetic restrictions and diagram statistics.
• Generating functions for small hook-lengths offer explicit formulas and enable asymptotic analysis, underpinning the enumeration of hooks in these restricted partitions.
• Monotonicity and bias phenomena have been established via combinatorial injections and analytic decompositions, offering deep insights into the structure of partition statistics.

A $t$-regular partition of an integer $n$ is a partition in which no part is divisible by $t$. The study of hook-lengths in $t$-regular partitions illuminates fine-grained combinatorial and analytic properties of restricted partition ensembles, with particular focus on the enumeration and bias phenomena related to hooks of given length across varying $t$. The central objects are the counts $b_{t,k}(n)$, representing the total number of hooks of length $k$ appearing in all $t$-regular partitions of $n$, and their generating functions. Recent breakthroughs have rigorously established monotonicity results and deep structural decompositions for the distribution of small hooks, notably confirming the Singh–Barman conjectures and revealing intricate injective and analytic frameworks underpinning these inequalities.

1. Definitions and Fundamental Objects

Given $t\ge2$, a partition $$\lambda = (\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_\ell > 0)$$ of $n$ is $t$-regular if $t\nmid \lambda_j$ for all $j$. Its Ferrers (Young) diagram is a left-justified grid of boxes corresponding to the parts, and each cell $(i,j)$ admits a hook of length $h(i,j) = \lambda_i - j + \lambda_j' - i + 1$, where $\lambda_j'$ is the conjugate partition. For fixed $k$, $b_{t,k}(n)$ denotes the aggregate number of hooks of length $k$ across all $t$-regular partitions of $n$, and its generating function is $B_{t,k}(q)=\sum_{n\ge0} b_{t,k}(n) q^n$ (Lin et al., 22 Oct 2025, Barman et al., 2024).

The sequence $\{b_{t,k}(n)\}_{n\ge0}$ encodes both arithmetic constraints (via $t$-regularity) and localized diagram statistics, serving as the primary analytic and combinatorial focus.

2. Generating Functions and Explicit Formulas

Closed-form expressions for $B_{t,k}(q)$ have been derived for small $k$, especially $k=1,2,3$, often involving infinite products and rational functions via $q$-Pochhammer symbols. For $k=1$, one has

$$\sum_{n\ge0}b_{t,1}(n)\,q^n = \frac{(q^t;q^t)_\infty}{(q;q)_\infty} \frac{q}{(1-q)(1-q^t)}$$

and for $k=2$ (Kim's formula): $$\sum_{n\ge0}b_{t,2}(n)\,q^n = \frac{(q^t;q^t)_\infty}{(q;q)_\infty} \left[ \frac{2q^2}{1-q^2} - \frac{q^t}{1-q^t} + \frac{q^{2t-1} - q^{2t} + q^{2t+1}}{1-q^{2t}} \right]$$ where $(a;q)_\infty = \prod_{k\ge0}(1-aq^k)$ (Lin et al., 22 Oct 2025, Lin et al., 30 Apr 2025). For $k=3$ analogous, albeit more complicated, expressions involve similar generating function structures.

These formulas enable analytic manipulation—subtraction, factorization, and decomposition—crucial for proving coefficientwise nonnegativity and for asymptotic analysis.

3. Inequalities, Biases, and Monotonicity Results

The foundational issue is the comparative behavior of $b_{t,k}(n)$ as either $t$ or $k$ varies. Major results include:

• For $t\ge3$ and all $n\ge0$,

$b_{t+1,2}(n) \ge b_{t,2}(n)$

as proved in full generality by Lin–Zang, confirming the Singh–Barman conjecture (Lin et al., 22 Oct 2025).

• For $t=3$, $b_{4,2}(n) \ge b_{3,2}(n)$ holds universally (Barman et al., 2024).
• For $t=2$, $b_{2,2}(n) \ge b_{2,1}(n)$ for $n>4$, and $b_{2,2}(n) \ge b_{2,3}(n)$ for all $n\ge0$ (Singh et al., 2024).
• Asymptotically, for fixed $t$, there exists $N=O(t^5)$ such that $b_{t,2}(n)\ge b_{t,1}(n)$ for all $n>N$ (Lin et al., 30 Apr 2025, Kim, 19 Jan 2025).
• For hooks of length 3, monotonicity with respect to $t$ also occurs: $b_{t+1,3}(n)\ge b_{t,3}(n)$ for all $n\ge0$ in computed cases (Singh et al., 2024, Saikia et al., 9 Jan 2026).

Failures of monotonicity have also been rigorously identified: in 2-regular partitions, for odd $k\ge3$, there exist infinitely many $n$ such that $b_{2,k}(n) < b_{2,k+1}(n)$, though for even $k$ greater than or equal to 4 monotonicity persists except at a single exceptional $n=k+1$ (Qu et al., 23 Jan 2025).

4. Proof Strategies and Structural Decompositions

The analytic mechanism involves decomposing generating function differences into sums of $q$-series, each corresponding to a disjoint combinatorial class of overpartitions or marked partitions. For example, in proving $b_{t+1,2}(n)\ge b_{t,2}(n)$, Lin–Zang decompose $\Delta_t(q)$ into six $q$-series, each the generating function of a well-defined family of overpartitions with congruence conditions on the overlined part (Lin et al., 22 Oct 2025). The six types correspond to:

Symbol	Overpartition Description	Congruence Condition
$a_t$	$(t+1)$-regular, unique overlined part even, not divisible by $t+1$	Even, $t+1\nmid$
$b_t$	$t$-regular, unique overlined part even, not divisible by $2t$	Even, $2t\nmid$
$c_t$	$t$-regular, unique overlined part odd multiple of $t$	Odd, $t\mid$
$d_t$	Overlined part even, divisible by $2$, not by $2(t+1)$, rest $t$-reg.	$2\mid$, $2(t+1)\nmid$
$e_t$	Exceptional overlined parts	$2t\pm1$
$f_t$	Exceptional overlined parts	$2t\pm3$

By constructing explicit injections between these combinatorial classes, negative contributions are compensated in the sums, guaranteeing nonnegativity of every coefficient.

In the even-$t$ case, the injections become further subdivided, breaking up $B\cup C$ into matches with duplicated $A$, etc. Reverse mappings are checked to ensure surjectivity and tightness.

For $t=3$ (the case $b_{4,2}(n)\ge b_{3,2}(n)$), the combinatorial proof employs a detailed block decomposition of parts modulo $12$, with explicit constructive injections and error term analysis, matched combinatorially with “orphan” partitions outside the image to reconcile any possible deficit in 2-hook counts (Barman et al., 2024).

5. Asymptotic Analysis and Analytical Frameworks

As $n\to\infty$, $t$-regular hook-counts exhibit growth governed by Meinardus-type formulas and circle-method asymptotics. The structure is: $$b_{t,k}(n) \sim D_{t,k} n^{-1/2} \exp\left(2\pi \sqrt{\frac{(1-1/t)n}{6}}\right)$$ where $D_{t,k}$ depends on $t$ and $k$ (Kim, 19 Jan 2025). For fixed $t$ and $k$ in a “square-root” regime, biases such as $b_{t,2}(n)\ge b_{t,1}(n)$ and $b_{t,2}(n)\ge b_{t,3}(n)$ hold for large $n$; the sign of bias is determined by the leading constants of the asymptotic expansion.

Comparison with $\ell$-distinct partitions reveals that

$$\lim_{n\to\infty} \frac{b_{\ell,k}(n)}{d_{\ell,k}(n)} = R_{\ell,k} < 1,$$

so there are asymptotically more $k$-hooks in the distinct class, but the ratio approaches $1$ as $\ell\to\infty$.

6. Algebraic and Combinatorial Frameworks: Littlewood Decomposition

The deeper algebraic structure relates $t$-regularity (t-core partitions) to Littlewood decomposition. Each partition decomposes into its $t$-core and $t$-quotient. Hooks divisible by $t$ correspond to the $t$-quotients, and in the $t$-core (i.e., strictly $t$-regular) case, the counts of hooks in residue classes modulo $t$ are explicitly expressed in terms of the quotient size data: $$|λ(0)|=0, \quad |λ(k)| + |λ(t-k)| = \sum_{(i,j)\in B_k} n_i n_j$$ where $$B_k = \{(i,j): 0\leq i<j\leq t-1, j-i\equiv\pm k\pmod t\}$$ (Dehaye et al., 2015).

This perspective connects hook statistics in the representation theory of $S_n$ via the Plancherel measure and polynomiality results.

7. Open Problems and Future Research Directions

Several questions remain open:

• Establish explicit closed-form generating functions for $b_{t,k}(n)$ for arbitrary $(t,k)$, especially $k\ge4$ (Singh et al., 2024).
• Characterize monotonicity thresholds $N_{t,k}$ for the bias $b_{t+1,k}(n)\ge b_{t,k}(n)$ and internal biases $b_{t,k}(n)\ge b_{t,k+1}(n)$, generalizing current polynomial bounds and asymptotic phenomena (Lin et al., 30 Apr 2025, Qu et al., 23 Jan 2025).
• Develop purely combinatorial (injection or sign-reversing involution) proofs for hook-length inequalities in higher lengths beyond $k=2$ (Lin et al., 22 Oct 2025).
• Analyze congruence properties and investigate partition-theoretic symmetries relating hook statistics in $t$-regular and $t$-distinct partitions.
• Extend analytic and combinatorial frameworks to other restricted partition families, e.g., self-conjugate or $t$-core partitions.

The synthesis of combinatorial injective techniques and analytic generating function methods provides a robust toolkit for approaching these problems. Continuing investigation in these directions promises further elucidation of the fine structure of partition statistics and their algebraic underpinnings.