Fixed-perimeter tipping-point conjecture for two residue classes

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.