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

TL;DR

This paper demonstrates that in an n-symbol alphabet, the count of words of length n with a number of distinct symbols significantly different from the expected (1-1/e)n decreases exponentially with n, using advanced combinatorial methods.

Contribution

It provides a new exponential decay bound for the number of words with atypical symbol diversity, employing Laplace's method and Stirling number bounds.

Findings

Number of words with atypical symbol counts decays exponentially.

Quantitative bounds expressed via inequalities.

Analysis based on Laplace's method and Stirling numbers.

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.