# Integer partitions with same divisors

Definitions: For $$\alpha < \beta \in (0,1)$$, let $$P(n,\alpha,\beta)$$ be the set of unordered integer partitions of $$n$$ where each part of the partition has size between $$n^\alpha$$ and $$n^\beta$$.

We a set of partitions, $$P$$, we define $$f(P)$$ to be the smallest integer $$k$$ such that for any distinct partitions $$x,y \in P$$, there exists $$1\le d \le k$$ such that the number of parts in $$x$$ divisible by $$d$$ differs from the number of parts in $$y$$ divisible by $$d$$. (note that as $$d$$ may equal $$1$$, we automatically handle cases where $$x,y$$ have different numbers of parts)

Questions: For $$\alpha,\beta$$, what can we say about the magnitude of $$g(n):= f(P(n,\alpha,\beta))$$? It is obvious that $$g(n) \le n^\beta$$. Can $$g$$ be subpolynomial?

For $$\alpha < \beta \in (0,1)$$, defining $$Q(n,\alpha,\beta)$$ to be the set of partitions with at most $$n^{1-\beta}$$ parts, each having size at least $$n^\alpha$$, can we get an upper bound of $$h(n):= f(Q(n,\alpha,\beta))$$?

Alternate Statement: It might make sense to instead define $$F(P)$$ to be the minimum integer $$k$$ such that for every pair of partitions $$x,y \in P$$, where $$x$$ and $$y$$ have zero parts of equal length, that there exists $$1\le d\le k$$ such that the number of parts in $$x$$ divisible by $$d$$ differs from the number of parts in $$y$$ divisible by $$d$$. We may then define $$G(n) = \max_{1\le m\le n}\{F(P(m,\alpha,\beta))\}, H(n) = \max_{1\le m\le n} \{F(Q(m,\alpha,\beta))\} .$$At least in my mind, this makes the asymptotics clearer and may be easier to handle as $$x,y$$ cannot be as similar as in the definition in $$f$$. While care is needed to convert bounds of $$G$$ and $$H$$ into bounds for $$g$$ and $$h$$, I would be quite interested in seeing answers to this case if it's easier.

Partial Progress: According to this post, the number of $$n^\alpha$$-smooth numbers less than $$n^\beta$$ is $$(1+o(1))\rho(\beta/\alpha)n$$. Thus, there exists a subset $$S \subset \{m \in \Bbb{N}:n^\alpha< m with positive density natural such that if there exists $$n+1$$ distinct partitions of $$n$$ only using parts which are elements of $$S$$, it follows that $$g(n) \ge n^\alpha$$.

I know that for a fixed subset of the naturals $$S$$ with positive natural density $$d>0$$, that the number of partitions of $$n$$ only using parts which are elements of $$S$$ is asymptotically $$e^{(1+o(1))C\sqrt{dn}}$$ where $$C = \pi\sqrt{2/3}$$. I do not know to control the asymptotics as $$n$$ and $$S$$ vary, but hopefully someone can manage to use this to show that $$g(n)\ge n^\alpha$$.