Search arXiv⌕ Search

arXiv subjects

Helmut Prodinger

Publications and source records attributed to Helmut Prodinger.

At least 19 recordsLinked to original sources
arXiv2026

The height of Dyck paths and checkerboard labellings

Dyck paths and certain black/white labelling of nodes leads to the \emph{white-height}. Using generating functions and tricks of the trade, we establish that the average white-height among Dyck paths of half length $n$ is asymptotic to $\frac12\sqrt{\pi n}$ for two different models. These are appealing results that could be presented to students to learn the trade. A last section links walks with double steps and 2-Motzkin paths. For them, the white-height is the natural concept. Folks who might be offended by the notion of \emph{white-height} might choose their own colours that they like.

math.CO↗
arXiv2026

The generating function of A348410 in OEIS using the diagonal method and another sequence (A001008) from OEIS

$a_n=[x^n](1-x)^{-n}(1-x^2)^{-n}$ is the sequence A348410 in the Encyclopedia of Integer Sequences. Using a method from Hautus and Klarner from 1971 and the software \textsf{Gfun} we find an algebraic equation for the generating function $g(z)=\sum_na_nz^n$. The second identity uses the \emph{generalized binomial series}, popularized in the textbook \emph{Concrete mathematics}.

math.CO↗
arXiv2026

Motzkin paths with two variants of level steps on odd levels -- a kernel method approach

The sequence A176677 in the Encyclopedia of Integer Sequences enumerates Motzkin paths where two types of horizontal steps may occur, but only on odd indexed levels. We show how to perform the enumeration, also dealing with partial such Motzkin paths leading to a particular level or to any level (open paths). The method is the kernel method where functional equations are manipulated in a suitable way. The coefficients of sequence A176677 satisfy a holonomic recursion that was recently discussed on the arxiv. We show how this can be established in an (almost) automatic fashion. Eventually we switch the roles of `odd' and `even'. One could also allow more versions of horizontal steps but we leave this to the interested readers.

math.CO↗
arXiv2026

Ordered trees with distinguished children

A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients can be done using the Lagrange inversion formula. Various parameters that are commonly studied for ordered trees can also be addressed here, like degree of the root, number of leaves, number of old leaves, height, height of leftmost leaf, and pathlength. We go through these instances and leave further parameters to later research, by either the author or some readers. Dealing with cubic equations is essential. Finally, ordered trees are replaced by marked ordered trees; they are then combined with the concept of distinguished children. Only the basic enumeration is provided at that stage.

math.CO↗
arXiv2026

The height of skew Dyck paths with two variants of downsteps

Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order $\sqrt n$.

math.CO↗
arXiv2025

Skew 2-Dyck paths via the kernel method

We continue on a recent concept introduced by Kariuki and Okoth, about skew 2-Dyck paths, introducing an additional down-step $L$, together with the usual steps $U$ (up) and $D$ down. There is the syntactical condition that $UL$ and $LU$ can never occur. An automaton that checks these conditions is introduced, and the relevant generating functions are obtained by applying the kernel method to three functional equations. It is briefly discussed how the setting can be extended to $t$-Dyck paths. As a benefit, prefixes of skew $t$-Dyck paths are also enumerated. An approach that scans 2-Dyck paths from right to left is also discussed.

math.CO↗
arXiv2025

Horton-Strahler numbers for binary butterfly trees: exact analysis

Peca suggested in a recent paper on the arxiv to consider binary butterfly trees and their Horton-Strahler numbers. The trees are obtained by glueing two binary trees together in a special way; the results are again binary trees but with a different probability distribution. A thorough combinatorial analysis is provided and leads asymptotically to the same results as for classical binary trees.

math.CO↗
arXiv2025

Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the $x$-axis and return to the $x$-axis. Versions where the return to the $x$-axis isn't required are also considered. A path is peakless (valleyless) if $UD$ (if $DU$) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of $UD$ resp., $DU$. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.

math.CO↗
arXiv2024

k-non-crossing trees and edge statistics modulo k

Instead of $k$-Dyck paths we consider the equivalent concept of $k$-non-crossing trees. This is our preferred approach relative to down-step statistics modulo $k$ (first studied by Heuberger, Selkirk, and Wagner by different methods). One symmetry argument about subtrees is needed and the rest goes along the lines of a paper by Flajolet and Noy.

math.CO↗
arXiv2024

Dispersed Dyck paths revisited

Dispersed Dyck paths are Dyck paths, with possible flat steps on level 0. We revisit and augment questions about them from the Encyclopedia of Integer Sequences, in a systematic way that uses generating functions and the kernel method.

math.CO↗
arXiv2023

Arndt-Carlitz compositions

Carlitz-compositions follow the restrictions of neighbouring parts $\sigma_{i-1}\neq\sigma_{i}$. The recently introduced Arndt-compositions have to satisfy $\sigma_{2i-1}>\sigma_{2i}$. The two concepts are combined to new and exciting objects that we call Arndt-Carlitz compositions.

math.CO↗
arXiv2023

Motzkin paths of bounded height with two forbidden contiguous subwords of length two

Motzkin excursions and meanders are revisited. This is considered in the context of forbidden patterns. Previous work by Asinowski, Banderier, Gittenberger, and Roitner is continued. Motzkin paths of bounded height are considered, leading to matrix equations and also to continued fractions. The enumeration is done by properly setting up bivariate generating functions which can be expanded using the kernel method.

math.CO↗