Bell number vs Factotial

Question

We have $B_n$ is Bell number and $n!$ - factorial. So, what is greater: $n!$ or $B_n$ ?

How it can be proven?

Answer 1

Factorials are bigger than Bell numbers, except for the initial cases when there is equality.

A comment from Emeric Deutsch on OEIS A048742 says that the difference counts

Number of permutations of $[n]$ which have at least one cycle that has at least one inversion when written with its smallest element in the first position. Example: $a(4)=9$ because we have $(1)(243)$, $(1432)$, $(142)(3)$, $(132)(4)$, $(1342)$, $(1423)$, $(1243)$, $(143)(2)$ and $(1324)$.

Since a count cannot be negative, and there is at least one example when $n \gt 2$, we need not look further.