Asymptotically half of binary words are shuffle squares

TL;DR

The paper proves that asymptotically half of all binary words of even length are shuffle squares, confirming a conjecture and providing an asymptotic count with high precision.

Contribution

It establishes the asymptotic proportion of binary shuffle squares among all binary words of even length, confirming a longstanding conjecture.

Findings

Number of binary shuffle squares of length 2n is approximately half of all binary words of that length.

Proves the conjecture that asymptotically half of binary words are shuffle squares.

Provides a precise asymptotic formula with an error term of o(n^{-1/15}).

Abstract

A binary shuffle square is a binary word of even length that can be partitioned into two disjoint, identical subwords. Huang, Nam, Thaper, and the first author conjectured that as $n\rightarrow \infty$, asymptotically half of all binary words of length $2n$ are shuffle squares. We prove this conjecture in a strong form, by showing that the number of binary shuffle squares of length $2n$ is $(\frac{1}{2} - o(n^{-1/15})) 2^{2n}$.