Shuffle Squares and Reverse Shuffle Squares
Xiaoyu He, Emily Huang, Ihyun Nam, Rishubh Thaper

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 be the family of {\it shuffle squares} in , words that can be partitioned into two disjoint identical subsequences. Let be the family of {\it reverse shuffle squares} in , 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 and 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 . We also prove a similar asymptotic formula for reverse shuffle squares that disproves their conjecture for . As these asymptotic formulas are vacuously true when the alphabet size is small, we…
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Taxonomy
Topicssemigroups and automata theory · Algorithms and Data Compression · Coding theory and cryptography
