Raney numbers, threshold sequences and Motzkin-like paths

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...

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Bibliographic Details
Published inDiscrete mathematics Vol. 345; no. 11; p. 113065
Main Author Rusu, Irena
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.11.2022
Subjects
Catalan numbers
Lattice paths
Motzkin paths
Raney numbers
Lattice paths
Motzkin paths
Catalan numbers
Raney numbers
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Summary: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.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2022.113065