Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $kn$ vertex balanced $k$-partite graph (each color is on $n$ vertices) to be $\mathcal G_{kn}$.
At every fixed $k\in\mathbb N$ there are many $m\in\mathcal N_{n}$ that do not have a $G\in \mathcal G_{kn}$ such that $f(G)=m$ (for $k=2$ refer Are all numbers from $1$ to $n!$ the number of perfect matchings of some bipartite graph?).
For a given $n$ and given $k$ denote by $\mathcal T_{k,n}$ to be the subset of $\mathcal N_{n}$ such that for every $m\in\mathcal T_{k,n}$ there is a $G\in\mathcal G_{kn}$ such that $f(G)=m$.
- Is the asymptotic $\log_2|\mathcal T_{k,n}|=\frac{\log_2|\mathcal N_n|}{\Omega((\log_2n)^{\frac1{k-1}})}$ at every $k\in\{2,3,\dots,\}$ valid?
- Is the asymptotic valid if $k=\Omega(\log_2\log_2n)$?
