Three Hoppy path problems and ternary paths

Deng, Eva Y. P.; Mansour, Toufik

doi:10.1016/j.dam.2007.08.015

Cited by 10 publications

(6 citation statements)

References 3 publications

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“…See also sequence A005494, which is 3 n C n .1; 1; 1=3; 1=3/. We now give a combinatorial proof showing that C n .a; b; c; d / given by (5) satisfies recurrence (1) using the interpretation given it in (4).…”

Section: Combinatorial Solutionmentioning

confidence: 91%

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A Recurrence Related to the Bell Numbers

Mansour

Shattuck

2012

Integers

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Section: Combinatorial Solutionmentioning

confidence: 91%

“…It is well known that there is no general procedure for solving recurrence relations, which is why some might say it is an art; see, for example, [4,8,9,17,18]. In this paper, we solve a seemingly new recurrence related to the Bell numbers involving a single index and four parameters given by where C 0 .a; b; c; d / D 1.…”

Section: Introductionmentioning

confidence: 99%

A Recurrence Related to the Bell Numbers

Mansour

Shattuck

2012

Integers

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“…Deng and Mansour [7] introduce a rabbit named Hoppy and let him move according to certain rules. While the story about Hobby is charming and entertaining, we do not need this here and move straight ahead to the enumeration issues.…”

Section: Hoppy Walksmentioning

confidence: 99%

A walk in my lattice path garden

Prodinger¹

2021

Preprint

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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 [47] (skew Dyck paths, Schröder 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 [17], 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 [6]. 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 max-turns and 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.

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“…Deng and Mansour [1] introduce a rabbit named Hoppy and let him move according to certain rules. At that stage, we don't need to know the rules.…”

Section: Hoppy Walksmentioning

confidence: 99%