Raney numbers, threshold sequences and Motzkin-like paths

Abstract

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up and down steps.Given three integers n,k,ℓ such that n≥1,k≥2 and 0≤ℓ≤k−2, a (k,ℓ)-threshold sequence of length n is any strictly increasing sequence S=(s1s2…sn) of integers such that ki≤si≤kn+ℓ. These sequences are in bijection with the (ℓ+1)-tuples of k-ary trees with a total of n internal nodes. We prove this result using counting arguments involving Raney numbers, but we also provide an explicit bijection between threshold sequences and tuples of trees. We further show how to represent threshold sequences as Motzkin-like paths with long up and down steps, and deduce that these paths are enumerated by the same Raney numbers. Finally, we illustrate the use of threshold sequences for finding new combinatorial identities.