Shuffle Squares and Reverse Shuffle Squares

Abstract

Let SS k (n) be the family of shuffle squares in [k] 2n , words that can be partitioned into two disjoint identical subsequences. Let RSS k (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 SS k (n) and RSS k (n) based on numerical evidence. We prove thatconfirming their conjecture for SS k (n). We also prove a similar asymptotic formula for reverse shuffle squares that disproves their conjecture for |RSS k (n)|. As these asymptotic formulas are vacuously true when the alphabet size is small, we study the binary case separately and prove that |SS 2 (n)| ≥ 2n n .

More Variations on Shuffle Squares

“…Even for the smallest binary case, we do not have a satisfactory answer. It was only recently proven by He, Huang, Nam, and Thaper [2] that the number of binary shuffle squares of length 2n is at least ( 2n n ), for n 3. An intriguing conjecture presented there states that almost every binary tangram is a shuffle square (in the sense that the probability of picking a shuffle square uniformly at random from all tangrams of length 2n tends to 1 with n tending to infinity).…”