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=(s_1 s_2 ... s_n) of integers such that ki <= s_i <= kn+l. These sequences are in bijection with ordered (l+1)-tuples of k-ary trees. We prove this result and identify the Raney numbers that count the (k,l)-threshold sequences. As a consequence, when k=2 and k=3, we deduce combinatorial identities involving Catalan numbers and powers of 2, and respectively Fuss-Catalan and Raney numbers. Finally, we 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.