This concerns difference/limit ratio results for special restricted partitions.

Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $b$ positive integers and each of them is at most $a$; if $r = 0$ and $a,b >0$, then $p(0,a,b)$ is taken to be one. Then, assuming $ab > 1$, $p(1,a,b) = 1$, and $p(ab,a,b) = 1$; moreover, $p(r,a,b) neq 0$ iff $0 leq r leq ab$. These are well known and closely related to Gauss polynomials (explicitly, the coefficient of $z^r$ in the $(a+b,b)$ Gaussian polynomial is $p(r,a,b)$) (notation varies considerably in the literature; I may have messed things up).

It is also well known that if $a, b$ are fixed, then the sequence $(p(0,a,b), p(1,a,b), dots , p(ab,a,b))$ is unimodal, and symmetric (about $ab/2$).

What I am interested in is the ratios of consecutive $p(r,a,m-a)$ with $r$ varying (note the change in notation, $b mapsto m-a$) over the interval $2a leq r leq a(m-a)/2$, with estimates uniform in $m$ and $a$, with $a$ not too close to $0$ or $m$ (for the latter, we can just restrict to $a leq m/2$).

To that end, define $$R_{m} = inf_{ 2a leq r leq a(m-a)/2; {rm and }3 leq a leq m-3} frac{p(r-1,a,m-a)}{p(r,a,m-a}.$$

It is important to keep track of the conditions below the infimum. Since $p(r, a,m-a)$ is increasing (as $r$ runs over $0,1,2, dots, a(m-a)/2$) (with $a$, $m$ fixed), obviously $R_{m} < 1$. It is relatively easy to show that $R_m geq 1/2$ for all but a few values of $m$.

**Question** Is it true that

$$R_{m} geq 1 – pmb oleft( 1right)$$

or something similar?

This might require a further restriction on $a$; small values (or those close to $m$) tend to screw it up; alternatively, impose more conditions on $r$, e.g., $r geq 3a$).

I imagine there are such results in the literature, but I am unfamiliar with it. For $r$ near the middle ($a(m-a)/2$), asymptotic estimates for $p(r,a,m-a)$ given in (RP Stanley & F Zanello, Some asymptotic results on q-binomial coefficients, Ann Comb 20 (2016) 623–34, Thm 2.6)) suggest that the problem probably reduces to relatively small $r$. An exact formula for $p(r,a,b)$ is given in (G Almkvist & GE Andrews, A Hardy-Ramanujan formula for restricted partitions, J Number Theory, 38 (1991) 135–144, Thm 4), but it is difficult to see how to use it. Without the $b$, asymptotic formulas were obtained by Szekeres (An asymptotic formula in the theory of partions I and II, Quart J Math 4 (1951) 85–108 and 4 (1953) 96–11).

*Motivation* I am looking at very simple random walks on the simplest torsion-free nonabelian group, given by $hg = z gh$ where $z$ is central. Then the coefficient of $z^r g^a h^b$ in $(g + h)^{a+b}$ is $p(r,a,b)$. I want to obtain the limit ratio result above (or some version of it), because it will almost certainly lead to a description of all extremal harmonic functions on the subcone starting at $1$ and proceeding by multiplication by $g+h$. Less cryptically, I have a bunch of such harmonic functions, and want to prove they constitute all of them.

In general, limit ratio results are easier to prove than establishing asymptotics (since the latter usually yields the former).