On the number of words with restrictions on the number of symbols

Abstract

We show that, in an alphabet of $n$ symbols, the number of words of length $n$ whose number of different symbols is away from $(1-1/e)n$, which is the value expected by the Poisson distribution, has exponential decay in $n$. We use Laplace's method for sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.

arXiv comment16 pages, 4 figures. In this second version, we correct a constant factor in Eq. (4) and we provide sharper bounds in Proposition 5 together with a more detailed proof