Finding binary words with a given number of subsequences

TL;DR

This paper explores the relationship between binary words with a specific number of subsequences and continued fractions, providing bounds on the length of such words and connecting to number theory conjectures.

Contribution

It establishes a novel connection between binary subsequence counts and continued fractions, deriving bounds on binary word lengths and linking to Zaremba's conjecture.

Findings

Binary words of length O(log n log log n) can have exactly n subsequences.

Under Zaremba's conjecture, this length bound improves to O(log n).

The work connects combinatorics on words with number theory and continued fractions.

Abstract

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this can be improved to $O(\log n)$ under assumption of Zaremba's conjecture.