Factorials $\pmod p$ and the average of modular mappings

Abstract: We have known that most sequences in $\mathcal{M}={1,2,\dots, M}$ with length $n$ will miss $Me^{-\lambda}$ of the total numbers of ${1,2,\dots,M}$ as the ratio $n/M$ tends to $\lambda$. Now we consider a more general case where the numbers in ${1,2,\dots,M}$ are achieved exactly k times by a 'random' sequence $f(1), f(2),\dots,f(n)$. We show that if $n/M\rightarrow \lambda$, then the limit has a Poisson distribution, that is, the proportion of sequences for which some number in $\mathcal{M}$ is achieved exactly $k$ times has the limit $\frac{\lambda^k}{k!}e^{-\lambda}$. We conjecture that this is the behavior of the factorial mapping modulo a prime and present a few supporting arguments.