A walk in my lattice path garden
Helmut Prodinger

TL;DR
This paper reviews various lattice path models, employing generating functions, bijections, and the kernel method, presenting new results and improvements on existing combinatorial enumeration problems, with extensive computational tools.
Contribution
It introduces new findings on Deutsch paths, amplitude, and improved analysis of Horton-Strahler numbers, integrating modern techniques like the slice method and kernel method.
Findings
New results on Deutsch paths in a strip
Improved analysis of Horton-Strahler numbers
Application of the slice technique and kernel method
Abstract
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are new, some have appeared before. The lattice path models we treated, are: Hoppy's walks, the combinatorics of sequence A002212 in \cite{OEIS} (skew Dyck paths, Schr\"oder paths, Hex-trees, decorated ordered trees, multi-edge trees, etc.) Weighted unary-binary trees also occur, and we could improve on our old paper on Horton-Strahler numbers \cite{FlPr86}, by using a different substitution. Some material on ternary trees appears as well, as on Motzkin numbers and paths (a model due to Retakh), and a new concept called amplitude that was found in \cite{irene}. Some new results on Deutsch paths in a strip are included as well. During the Covid period, I…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Stochastic processes and statistical mechanics · Data Management and Algorithms
