Singh–Barman Conjectures

• Singh–Barman conjectures are statements about inequalities among hook-length statistics b_{t,k}(n) in t-regular partitions, emphasizing positivity and monotonicity.
• They leverage q-series generating functions, combinatorial bijections, and block decomposition to analyze the distribution of hook lengths in partitions.
• Recent proofs confirm the 3-regular bias and establish monotonicity in 2-regular cases, highlighting refined combinatorial techniques and asymptotic verifications.

The Singh–Barman conjectures concern inequalities among the hook-length statistics $b_{t,k}(n)$ in $t$-regular partitions, where $b_{t,k}(n)$ is the number of hooks of length $k$ across all $t$-regular partitions of an integer $n$. These conjectures—focusing on positivity and monotonicity of specific hook-length biases—have catalyzed sustained investigation into the fine structure of $t$-regular partition statistics, combinatorial bijections, and generating function techniques. This article surveys the conjectures, their statements, resolution, and ramifications in the context of recent research.

1. Definition of $t$-Regular Hook-Length Statistics

A partition $$\lambda=(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r)$$ of $n$ is $t$-regular if no part $\lambda_i$ is divisible by $t$. The Ferrers or Young diagram of $\lambda$ comprises rows of boxes, with $\lambda_i$ boxes in row $i$. For each box $(i,j)$, its hook comprises all boxes to its right within the same row, all boxes below it in the same column, and the box itself—yielding the hook-length

$h_{i,j} = (\lambda_i - j) + (\lambda'_j - i) + 1,$

where $\lambda'$ is the conjugate partition. For $b_{t,k}(n)$, all $t$-regular partitions $\lambda$ of $n$ are considered; $b_{t,k}(n)$ is the total number of hook-length-$k$ boxes summed across $\mathcal B_t(n)$, the set of all $t$-regular partitions of $n$ (Barman et al., 2024).

The generating function for $t$-regular partitions (distinct from the hook-length generating function) is

$$\sum_{n=0}^{\infty} b_t(n)\,q^n = \frac{(q^t;q^t)_\infty}{(q;q)_\infty},$$

where $(a;q)_\infty = \prod_{m=0}^{\infty}(1-aq^m)$. For hook-lengths (notably $k=2$), explicit $q$-series identities have been derived using $q$-binomial coefficients (Qu et al., 23 Jan 2025, Barman et al., 2024).

2. Formulation of the Singh–Barman Conjectures

Singh and Barman introduced two main conjectures relating hook-length biases in $t$-regular partitions (Qu et al., 23 Jan 2025):

1. Hook-Length Bias for $3$-Regular Partitions

$$\forall n\geq 28,\qquad b_{3,2}(n) - b_{3,1}(n) > 0,$$

or equivalently,

$b_{3,2}(n) \geq b_{3,1}(n)\qquad (n \geq 28).$

This asserts dominance of $2$-hooks over $1$-hooks for "sufficiently large" $n$ in $3$-regular partitions.

2. Monotonicity in Hook-Length Biases for $2$-Regular Partitions

$$\forall k\geq 3,\quad \forall n \geq 0,\; n \neq k+1,\qquad b_{2,k}(n) \geq b_{2,k+1}(n).$$

This posits that for $k \geq 3$ and $n \neq k+1$, the number of $k$-hooks is at least the number of $(k+1)$-hooks in 2-regular partitions of $n$.

A third conjecture emerged from empirical evidence (Qu et al., 23 Jan 2025):

$$\text{For all even } k \geq 8,\;\forall n \geq 0,\;n\neq k+1,\qquad b_{2,k}(n) \geq b_{2,k+1}(n).$$

This refines Conjecture 2 to cases $k=4,6$ and even $k\geq 8$, excluding a single small anomaly at $n=k+1$.

3. Resolution and Proofs

First Conjecture: Bias in $3$-Regular Partitions

The proof of $b_{3,2}(n) \geq b_{3,1}(n)$ for $n\geq 28$ rests on generating-function manipulations and careful partition combinatorics (Qu et al., 23 Jan 2025). The exact generating function for $\Delta(n) := b_{3,2}(n) - b_{3,1}(n)$ is developed: $$\sum_{n \geq 0} \Delta(n) q^n = -q(1+q^2+q^4)(1+q^3+q^6)\left(U(q) - q^3 V(q)\right),$$ where $U(q)$ and $V(q)$ have combinatorial interpretations as generating functions for special triples of partitions $(\alpha,\beta,\gamma)$ defined by congruence and minimal part constraints.

The approach partitions the relevant sets into seven disjoint classes, constructs explicit bijections $\phi_i$ for $i=1,\ldots,6$ between corresponding classes, and bounds cardinalities via combinatorial estimates for the seventh. Large-$n$ behavior is rigorously established ($n\geq152$), with finite verification for intermediate $n$ ($28\leq n\leq162$), resulting in full confirmation of Conjecture 1.

Second Conjecture: Monotonicity for $2$-Regular Partitions

Combinatorial and analytic techniques demonstrate a dichotomy depending on the parity of $k$ (Qu et al., 23 Jan 2025). For odd $k\geq 3$, generating function manipulations using rational functions (Craig–Dawsey–Han formula) yield, for $k=2t+1$,

$$\frac{1-q}{(-q;q)_\infty\, \sum_{n\geq0}(b_{2,2t+1}(n)-b_{2,2t+2}(n))q^n} = \frac{A_t(q)}{B_t(q)},$$

with $A_t(q)$ accumulating negative coefficient sums for large $n$. This implies that $b_{2,k}(n) < b_{2,k+1}(n)$ infinitely often for odd $k$, refuting monotonicity in this setting.

In contrast, for $k=4$, $6$, explicit $q$-series expansions and positivity lemmas verify $b_{2,4}(n) \geq b_{2,5}(n)$ except $n=5$, and $b_{2,6}(n) \geq b_{2,7}(n)$ except $n=7$. Empirical evidence for even $k\geq8$ supports a refined monotonicity conjecture (Conjecture 3), suggesting the original is correctly restricted to these values.

4. Generalizations and Related Results

A third principal direction posited by Singh, Barman, Mahanta, and others involves comparing $b_{t+1,2}(n)$ to $b_{t,2}(n)$ for $t\geq 3$—i.e., monotonicity of $2$-hook statistics across increasing $t$ (Barman et al., 2024, Lin et al., 22 Oct 2025). Initial proofs for $t=3$ use explicit block decomposition mod 12, constructing a map $\Phi_{3,n}$ from $3$-regular to $4$-regular partitions preserving or compensating 2-hooks, and refining with a secondary map $\Psi$ for core cases.

In (Lin et al., 22 Oct 2025), the full general case is resolved: for all $t \geq 3$ and $n \geq 0$,

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

barring the unique exception $(t,n)=(2,3)$. The proof utilizes generating-function decomposition into six $q$-series, each counting certain classes of one-overlined-part overpartitions ("OPO-overpartitions"). Negative classes are paired injectively with positive classes, ensuring nonnegativity of the difference for all $n$.

The approaches are combinatorial and constructive, with explicit injections defined for both odd and even $t$, uniformly covering all $t$.

5. Methods and Technical Frameworks

Key methodologies include:

• Generating-Function Analysis:

Explicit calculations with $q$-series, $q$-Pochhammer symbols, and rational generating functions for $b_{t,k}(n)$.

• Block Decomposition:

Partitioning $t$-regular partitions into blocks based on congruence classes modulo $t(t+1)$ or similar, permitting localized analysis and map constructions.

• Combinatorial Bijections and Injections:

Explicit design of maps maintaining or offsetting hook counts between partition families (notably $\Phi_{t,n}$, $\Psi$ and various $\phi_i$, $\zeta_j$ in the OPO-overpartition context).

• Asymptotic and Computational Verification:

Positivity of certain rational functions and empirical verification for small $n$ supplement combinatorial arguments.

6. Implications and Open Problems

These results establish a nuanced landscape of hook-length bias and monotonicity in $t$-regular partitions. The Singh–Barman conjectures hold unconditionally for the $3$-regular positivity and for monotonicity in $2$-hook statistics across $t$, with limitations for certain odd $k$ values in $2$-regular monotonicity. Empirical evidence and refined conjectures indicate robust behavior for even $k$, with no known exceptions beyond the established anomalies.

A plausible implication is that deeper bias phenomena in $t$-regular partitions, potentially for other hook-lengths $k$ or for asymptotic regimes, are accessible via the combinatorial partitioning and injection framework exemplified in recent proofs. The possible connections to Nekrasov–Okounkov expansions and modular-form interpretations of partition statistics remain open directions for research (Lin et al., 22 Oct 2025). The complexity of extending block decomposition and bijective compensation methods to arbitrary $t$ and $k$ suggests the need for further advances in both combinatorial and analytic techniques.

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Singh–Barman Conjectures: Summary Table

Conjecture	Statement	Status
1. $3$-regular bias	$b_{3,2}(n) \geq b_{3,1}(n)$ for $n \geq 28$	Proven (Qu et al., 23 Jan 2025)
2. $2$-regular monotonicity	$b_{2,k}(n) \geq b_{2,k+1}(n)$ for $k\geq 3$, $n\neq k+1$	True for $k$ even (except $n=k+1$); false for $k$ odd (Qu et al., 23 Jan 2025)
3. $2$-hook monotonicity in $t$	$b_{t+1,2}(n) \geq b_{t,2}(n)$ for $t\geq3$, all $n$	Proven (Lin et al., 22 Oct 2025)

The continuing exploration of hook-length biases in $t$-regular partitions contributes to a deeper understanding of partition arithmetic and combinatorial structures within number theory.