Compacted binary trees admit a stretched exponential

Progressive and Rushed Dyck Paths

“…The estimate (8) and the limit law (9) are directly in [9]. The limit law (10) is also derived using a local limit law found in the same reference.…”

“…Their enumerating sequence exhibit a remarkable asymptotic estimate of the form 4 n e −νn 1/3 n −5/6 . This sort of stretched exponential has attracted recent attention in many combinatorial contexts [11,6,10,8], but it highly unusual for lattice paths, which usually have a form µ n n γ , with γ normally 0, −1/2 or −3/2 in one dimension [2]. Culminating paths, for their part, asymptotically number 2 n /(4n) [4,Proposition 4.1].…”

On the critical exponents of generalized ballot sequences in three dimensions and large tandem walks

“…where C is a constant, µ the exponential growth, and α the critical exponent. Note that general sequences may include additional terms such as n!, n n , or more complicated terms such as µ n σ known as stretched exponentials [11]. In the context of this note, the structure (2) will suffice.…”