Progressive and Rushed Dyck Paths
For combinatorialists, this resolves an open enumerative equivalence and introduces new connections between lattice paths and trees, though the result is incremental.
The authors present a bijection proving that two families of Dyck paths (progressive and rushed) have the same enumeration sequence (A287709). They also connect rushed paths to one-sided trees, which have asymptotic enumeration involving a stretched exponential, and suggest other related structures may have similar properties.
We call progressive paths and rushed paths two families of Dyck paths studied by Asinowski and Jelinek, which have the same enumerating sequence (OEIS entry A287709). We present a bijection proving this fact. Rushed paths turn out to be in bijection with one-sided trees, introduced by Durhuus and Unel, which have an asymptotic enumeration involving a stretched exponential. We conclude by presenting several other classes of related lattice paths and directed animals that may have similar asymptotic properties.