A walk in my lattice path garden

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 spent much time with this beautiful concept that I dare to call Deutsch paths, since Emeric Deutsch stands at the beginning with a problem that he posted in the American Mathematical Monthly some 20 years ago. Peaks and valleys, studied by Rainer Kemp 40 years under the names \textsc{max}-turns and \textsc{min}-turns, are revisited with a more modern approach, streamlining the analysis, relying on the `subcritical case' (named so by Philippe Flajolet), the adding a new slice technique and once again the kernel method.