Constructions for Clumps Statistics

We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This parameter has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We emphasize the fact that, in contrast with the probabilistic approach which only provides asymptotic results, the combinatorial approach provides exact results that are useful when considering short sequences.

💡 Research Summary

The paper addresses the problem of counting “clumps” – maximal, non‑overlapping sets of occurrences of a given word or a finite set of words – in random texts. While earlier work (Schbath, Reinert, etc.) studied clumps using probabilistic tools such as Poisson and compound‑Poisson approximations via the Chen‑Stein method, those approaches yield only asymptotic results and become inaccurate for short sequences. The authors fill this gap by developing a fully combinatorial framework that provides exact generating functions for a variety of clump‑related statistics.

The authors start by recalling the Régnier‑Szpankowski language decomposition, which splits a text into four languages: Right (R), Minimal (M), Ultimate (U), and Not (N). For a single word w (or a reduced set of words U) these languages describe, respectively, the prefix up to the first occurrence, the minimal bridge between two occurrences, the suffix after the last occurrence, and the set of texts containing no occurrence. By introducing the autocorrelation set C of a word (the set of all non‑empty proper suffixes that are also prefixes), they define C◦ = C \ {ε} and construct a prefix code K that uniquely generates the Kleene star C*.

Lemma 1 proves that K indeed yields an unambiguous decomposition of C*. Lemma 2 shows that K is contained in the Minimal language M and that the difference M \ K can be expressed as L·w for some non‑empty language L. Lemma 3 then gives the fundamental decomposition of any text as
N ∪ R·w·(C*)·