For positive integers d, define h_d^(a)(n) as the number of partitions of n into parts that are d-distinct and at least a, and define ell_m^(m_1,m_2)(n) as the number of partitions of n into parts congruent to m_1 or m_2 modulo m. For positive integers 0 2d+2, -∞ & m<2d+2. This conjecture proposes a sharp asymptotic tipping point at m=2d+2, motivated by numerical evidence and by known comparisons between fixed-perimeter partition functions. The supplied statement does not assert a result in the boundary case m=2d+2.

Fixed-perimeter tipping-point conjecture for two residue classes

For positive integers $d$ , define $h_d^{(a)}(n)$ as the number of partitions of $n$ into parts that are $d$ -distinct and at least $a$ , and define $\ell_m^{(m_1,m_2)}(n)$ as the number of partitions of $n$ into parts congruent to $m_1$ or $m_2$ modulo $m$ . For positive integers $0<m_1<m_2\leq m$ and $0<a\leq d$ , the fixed-perimeter tipping-point conjecture.

\lim_{n\to\infty}\left(h_d^{(a)}(n)-\ell_m^{(m_1,m_2)}(n)\right)= \begin{cases} +\infty & m>2d+2,\\ -\infty & m<2d+2. \end{cases}

This conjecture proposes a sharp asymptotic tipping point at $m=2d+2$ , motivated by numerical evidence and by known comparisons between fixed-perimeter partition functions. The supplied statement does not assert a result in the boundary case $m=2d+2$ .

References

Primary source

Gabriel Gray, Emily Payne, Holly Swisher and Ren Watson, “Fixed perimeter analogues of some partition results”, arXiv:2502.12394 (2025).

Additional references

2 papers in this index state this conjecture (2025). The statement above is taken from the most recent of them; the others are arXiv:2502.11929.

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Progress summary

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Open

The conjecture remains open: numerical evidence supports the proposed cutoff, but no proof or counterexample has been publicly reported.

The conjecture appears as Conjecture $1.5$ in a preprint dated February 2025, predicting opposite eventual dominance on the two sides of the threshold $m=2d+2$ . The boundary case is explicitly left more complicated and unresolved apart from selected cases.

Known results

Straub, 2016: established the fixed-perimeter identity relating odd-part and distinct-part partitions to Fibonacci numbers.
Fu and Tang, 2018: generalized Straub’s identity.
Chen et al., 2024: refined these results for corresponding $d$ -distinct and residue-class counting functions.
Kang and Kim, 2021: proved the analogous tipping-point theorem for ordinary fixed-size partition functions.

February 2025 status

The authors present the strict-inequality assertions as conjectural and supported by experimentation, not as proved theorems. Proposition $1.6$ settles selected boundary comparisons, while other boundary regimes remain unclear; no later source reports a proof, counterexample, or verification.

Current status (as of August 2026): The conjecture remains open; only selected boundary-case comparisons are proved, and neither strict-inequality assertion has a reported proof or counterexample.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

Full proof, with exact exponential asymptotics. Let $s=d+1$ , $u=m_1$ , and $v=m_2$ . The source's proved generating functions (equations (4.4) and (5.2)) give

H(z)=\sum_{n\ge1}h_d^{(a)}(n)z^n=\frac{z^a}{1-z-z^s},\qquad L(z)=\sum_{n\ge1}\ell_m^{(u,v)}(n)z^n =\frac{z^u(1-z)+z^v}{(1-z)^2-z^m}.

For each real $t>0$ , denote by $\rho_t\in(0,1)$ the unique solution of $1-\rho_t=\rho_t^t$ . The map $t\mapsto\rho_t$ is strictly increasing: if $t_2>t_1$ , then

1-\rho_{t_1}-\rho_{t_1}^{t_2} =\rho_{t_1}^{t_1}-\rho_{t_1}^{t_2}>0.

The unique dominant pole of $H$ is $\rho_s$ . Indeed, every zero $z$ of $1-z-z^s$ with $|z|\le\rho_s$ satisfies

1=|z+z^s|\le |z|+|z|^s\le\rho_s+\rho_s^s=1,

and equality forces $z=\rho_s$ . The pole is simple.

Similarly, writing $\tau=\rho_{m/2}$ , every zero of $(1-z)^2-z^m$ with $|z|\le\tau$ satisfies

1=\left|\frac{z^m}{(1-z)^2}\right| \le\frac{|z|^m}{(1-|z|)^2} \le\frac{\tau^m}{(1-\tau)^2}=1.

Equality in $|1-z|\ge1-|z|$ forces $z=\tau$ . This pole is simple, and its numerator is strictly positive. Partial fractions therefore give

h_d^{(a)}(n)\sim \frac{\rho_s^{a-1}}{1+s\rho_s^{s-1}}\rho_s^{-n}, \quad \ell_m^{(u,v)}(n)\sim \frac{\tau^u(1-\tau)+\tau^v} {\tau\left(2(1-\tau)+m\tau^{m-1}\right)} \tau^{-n}.

Both leading constants are positive. If $m>2s$ , strict monotonicity gives $\rho_s<\tau$ , so $h_d^{(a)}(n)-\ell_m^{(u,v)}(n)\sim C_H\rho_s^{-n}\to+\infty$ . If $m<2s$ , then $\tau<\rho_s$ , so the difference is asymptotic to $-C_L\tau^{-n}\to-\infty$ . Since $2s=2d+2$ , this proves the conjecture for every permitted $d,a,m,u,v$ .

Additional critical-case information. When $m=2s$ , the common denominator factors as

(1-z)^2-z^{2s}=(1-z-z^s)(1-z+z^s).

At $\rho=\rho_s$ , the numerator of $H-L$ equals

J=2\rho^{a+s}-\rho^{u+s}-\rho^v.

Whenever $J\ne0$ ,

h_d^{(a)}(n)-\ell_{2s}^{(u,v)}(n) \sim \frac{J}{2\rho^{s+1}(1+s\rho^{s-1})}\rho^{-n},

so its sign is determined explicitly by $J$ ; the choice $(u,v)=(a,a+s)$ gives exact equality for every $n$ .

Source: Gray, Payne, Swisher, and Watson, Fixed perimeter analogues of some partition results, Discrete Mathematics 349 (2026), 114968, Conjecture 1.5.