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Helmut Prodinger

Publications and source records attributed to Helmut Prodinger.

At least 37 records · Page 2Linked to original sources
arXiv2023

MIN-turns and MAX-turns in k-Dyck paths: a pure generating function approach

$k$-Dyck paths differ from ordinary Dyck paths by using an up-step of length $k$. We analyze at which level the path is after the $s$-th up-step and before the $(s+1)$st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago the terms \textsc{max}-terms and \textsc{min}-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.

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arXiv2023

Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection

A well-known bijection between Motzkin paths and ordered trees with outdegree always $\le2$, is lifted to Grand Motzkin paths (the nonnegativity is dropped) and an ordered list of an odd number of such $\{0,1,2\}$ trees. This offers an alternative to a recent paper by Rocha and Pereira Spreafico.

math.CO↗
arXiv2023

Peakless Motzkin paths of bounded height

There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a strip $[0..\ell]$, and as consequence the asymptotics of the average height, which is given by $2\cdot 5^{-1/4}\sqrt{πn}$. Methods include the kernel method and singularity analysis of generating functions.

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arXiv2023

S-Motzkin paths with catastrophes and air pockets

So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.

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arXiv2022

Enumeration of partial Lukasiewicz paths

Łukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length prefixes (resp. suffixes) of these paths ending at height $k\geq 0$ with a given type of step. We make a similar study for prefixes of height at most $t\geq 0$. Using the explicit forms for the paths of bounded height, we evaluate the average height asymptotically. For fixed $k$ and $n\to\infty$, this quantity behaves as $\sqrt{πn}$. Finally we study (in the same way) prefixes of alternate Łukasiewicz paths, i.e., Łukasiewicz paths that do contain two consecutive steps with the same direction.

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arXiv2022

Partial skew Dyck paths -- a kernel method approach

Skew Dyck are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end with decorated Dyck paths. We analyze partial versions of them where the path ends on a fixed level $j$, not necessarily at level 0. We exclusively use generating functions and derive them with the celebrated kernel method. In the second part of the paper, a dual version is studied, where the paths are read from right to left. In this way, we have two types of up-steps, not two types of down-steps, as before. A last section deals with the variation that the negative territory (below the $x$-axis) is also allowed. Surprisingly, this is more involved in terms of computations.

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arXiv2022

Partial Skew Motzkin Paths

Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left step $(-1,-1)$ is additionally allowed, but the path is not allowed to intersect itself. The enumeration of these combinatorial objects was known \cite{Qing}; here, using the kernel method, we extend the results by allowing them to end at a prescribed level $j$. The approach is completely based on generating functions. Asymptotics of the total number of objects as well as the average height are also given.

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arXiv2022

Skew Dyck paths without up--down--left

Skew Dyck paths without up-down-left are enumerated. In a second step, the number of contiguous subwords 'up-down-left' are counted. This explains and extends results that were posted in the Encyclopedia of Integer Sequences.

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arXiv2022

Partial Dyck paths with Air Pockets

Dyck paths with air pockets are obtained from ordinary Dyck paths by compressing maximal runs of down-steps into giant down-steps of arbitrary size. Using the kernel method, we consider partial Dyck paths with air pockets, both, from left to right and from right to left.

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arXiv2022

Knödel walks in a Böhm-Hornik environment

Ideas of Knödel and Böhm-Hornik about walks in certain graphs, resembling the classical symmetric random walk on the integers, are combined. All the relevant generating functions (although occasionally quite involved) are made fully explicit.

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arXiv2022

Deepest nodes in marked ordered trees

A variation of ordered trees, where each rightmost edge might be marked or not, if it does not lead to an endnode, is investigated. These marked ordered trees were introduced by E. Deutsch et al.\ to model skew Dyck paths. We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches $\frac53$ when the number of nodes gets large. This is to be compared to standard ordered trees where the average number of deepest nodes approaches $2$.

math.CO↗
arXiv2022

A walk in my lattice path garden

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, 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 \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.

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arXiv2022

Skew Dyck paths with catastrophes

Skew Dyck paths are like Dyck paths, but an additional south-west step $(-1,-1)$ is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used.

math.CO↗
arXiv2022

Skew Dyck paths having no peaks at level 1

Skew Dyck paths are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end up with decorated Dyck paths. Sequence A128723 of the Encyclopedia of Integer Sequences considers such paths where peaks at level 1 are forbidden. We provide a thorough analysis of a more general scenario, namely partial decorated Dyck paths, ending on a prescribed level $j$, both from left-to-right and from right-to-left (decorated Dyck paths are not symmetric). The approach is completely based on generating functions.

math.CO↗
arXiv2021

An online bin-packing problem with an underlying ternary structure

Following an orginal idea by Knödel, an online bin-packing problem is considered where the the large items arrive in double-packs. The dual problem where the small items arrive in double-packs is also considered. The enumerations have a ternary random walk flavour, and for the enumeration, the kernel method is employed.

math.CO↗
arXiv2021

Lattice paths with infinitely many down steps -- the negative boundary model

We consider a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ down-steps $(1,-j)$, for $j\ge2$ are allowed. We give credits to Emeric Deutsch for that. The enumeration of such objects living in a strip is performed. Methods are the kernel method and techniques from linear algebra.

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