Shuffle squares and reverse shuffle squares

Abstract

Let SSk(n) be the family of shuffle squares in [k]2n, words that can be partitioned into two disjoint identical subsequences. Let RSSk(n) be the family of 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 SSk(n) and RSSk(n) based on numerical evidence. We prove that |SSk(n)|=1n+12nnkn−2n−1n+1kn−1+On(kn−2),confirming their conjecture for |SSk(n)|. We also prove a similar asymptotic formula for reverse shuffle squares that disproves their conjecture for |RSSk(n)|. As these asymptotic formulas are vacuously true when the alphabet size is small, we study the binary case separately and prove that |SS2(n)|≥2nn.