Optimality of the partition-matching exponent for set-partition avoidance

Let $\pi$ be a set partition. Denote by $B_n(\pi)$ the number of set partitions of $[n]$ avoiding $\pi$, and let $\operatorname{pm}(\pi)$ be the partition-matching number of $\pi$. Then the following bounds are conjectured to hold.

Partition-matching exponent conjecture.

• Weak form:

$$\lim_{n\to\infty}\frac{\log B_n(\pi)}{n\log n}=1-\frac{1}{\operatorname{pm}(\pi)}.$$

• Strong form: There exists a constant $c_2(\pi)$ such that

$$B_n(\pi)\leq c_2(\pi)^n n^{n\left(1-\frac{1}{\operatorname{pm}(\pi)}\right)}$$

for all $n$.

The lower bound with the same exponent is proved in the paper, while the conjecture concerns the matching upper bounds. The strong form implies the weak form when combined with the established lower bound; both are motivated by the failure of Alweiss's earlier exponent involving $\operatorname{th}(\pi)$ when $\operatorname{pm}(\pi)>\operatorname{th}(\pi)$.