Counting Lyndon Subsequences

TL;DR

This paper investigates the enumeration of Lyndon subsequences within strings, establishing maximum and expected counts, and analyzing the diversity of such subsequences to deepen understanding in string combinatorics.

Contribution

It introduces the problem of counting Lyndon subsequences and provides results on maximum, expected, and distinct counts, advancing the theoretical understanding of Lyndon structures.

Findings

Maximum total number of Lyndon subsequences in a string

Expected total number of Lyndon subsequences in a string

Expected number of distinct Lyndon subsequences

Abstract

Counting substrings/subsequences that preserve some property (e.g., palindromes, squares) is an important mathematical interest in stringology. Recently, Glen et al. studied the number of Lyndon factors in a string. A string $w = uv$ is called a Lyndon word if it is the lexicographically smallest among all of its conjugates $vu$. In this paper, we consider a more general problem "counting Lyndon subsequences". We show (1) the maximum total number of Lyndon subsequences in a string, (2) the expected total number of Lyndon subsequences in a string, (3) the expected number of distinct Lyndon subsequences in a string.