Maximal Number of Subword Occurrences in a Word

Fang, Wenjie

doi:10.4230/LIPIcs.AofA.2024.3

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LIPIcs.AofA.2024.3.pdf

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Document Identifiers

DOI: 10.4230/LIPIcs.AofA.2024.3
URN: urn:nbn:de:0030-drops-204387

Author Details

Wenjie Fang

Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France

Acknowledgements

I would like to thank Stéphane Vialette for bringing the question of maximal number of subword occurrences of a given word to our attention, and for giving an idea for Proposition 3.4.

Cite AsGet BibTex

Wenjie Fang. Maximal Number of Subword Occurrences in a Word. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.3

Abstract

We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We then give upper and lower bounds of minimal subword entropy for words of fixed length in a fixed alphabet, and also showing that minimal subword entropy per letter has a limit value. A better upper bound of minimal subword entropy for a binary alphabet is then given by looking at certain families of periodic words. We also give some conjectures based on experimental observations.

Subject Classification

ACM Subject Classification

Mathematics of computing → Enumeration
Mathematics of computing → Combinatorics on words

Keywords

Subword occurrence
subword entropy
enumeration
periodic words

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