Shuffle Squares and Reverse Shuffle Squares

TL;DR

This paper derives asymptotic formulas for the number of shuffle squares and reverse shuffle squares in words over an alphabet, confirming a conjecture for shuffle squares and disproving it for reverse shuffle squares, with special results for binary alphabets.

Contribution

It proves the conjectured asymptotic formula for shuffle squares and provides a counterexample for reverse shuffle squares, advancing understanding of these combinatorial structures.

Findings

Confirmed the asymptotic formula for shuffle squares.

Disproved the conjectured formula for reverse shuffle squares.

Established a lower bound for binary alphabet shuffle squares.

Abstract

Let $\mathcal{SS}_k(n)$ be the family of {\it shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint identical subsequences. Let $\mathcal{RSS}_k(n)$ be the family of {\it reverse shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint subsequences which are reverses of each other. Henshall, Rampersad, and Shallit conjectured asymptotic formulas for the sizes of $\mathcal{SS}_k(n)$ and $\mathcal{RSS}_k(n)$ based on numerical evidence. We prove that \[ \lvert \mathcal{SS}_k(n) \rvert=\dfrac{1}{n+1}\dbinom{2n}{n}k^n-\dbinom{2n-1}{n+1}k^{n-1}+O_n(k^{n-2}), \] confirming their conjecture for $\mathcal{SS}_k(n)$. We also prove a similar asymptotic formula for reverse shuffle squares that disproves their conjecture for $\lvert \mathcal{RSS}_k(n) \rvert$. As these asymptotic formulas are vacuously true when the alphabet size is small, we study the binary case separately and prove that $|\mathcal{SS}_2(n)| \ge \binom{2n}{n}$.