Helmut Prodinger
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View article: Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs
Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs Open
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 D…
View article: Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs
Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs Open
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 $UD$ …
View article: Grand Motzkin Paths and \(\{0,1,2\}\)-Trees -- A Simple Bijection
Grand Motzkin Paths and \(\{0,1,2\}\)-Trees -- A Simple Bijection Open
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…
View article: Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences
Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences Open
Descents of odd length in Dyck paths are discussed, taking care of some variations. The approach is based on generating functions and the kernel method and augments relations about them from the Encyclopedia of Integer Sequences, that were…
View article: MIN-turns and MAX-turns in k-Dyck paths: A pure generating function approach
MIN-turns and MAX-turns in k-Dyck paths: A pure generating function approach Open
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…
View article: On \({k}\)-Dyck Paths with a Negative Boundary
On \({k}\)-Dyck Paths with a Negative Boundary Open
Paths that consist of up-steps of one unit and down-steps of \(k\) units, being bounded below by a horizontal line \(-t\), behave like \(t+1\) ordered tuples of \(k\)-Dyck paths, provided that \(t\le k\). We describe the general case, allo…
View article: k-non-crossing trees and edge statistics modulo k
k-non-crossing trees and edge statistics modulo k Open
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 …
View article: Dispersed Dyck paths revisited
Dispersed Dyck paths revisited Open
Dispersed Dyck paths are Dyck paths with possible flat steps on level 0. Questions about dispersed Dyck paths, from the Encyclopedia of Integer Sequences, are revisited and augmented in a systematic way that uses generating functions and t…
View article: Dispersed Dyck paths revisited
Dispersed Dyck paths revisited Open
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.
View article: Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths
Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths Open
Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A mor…
View article: Prefixes of Stanley’s Catalan Paths with Odd Returns to the \(x\)-axis–Standard Version and Skew Catalan-Stanley Paths
Prefixes of Stanley’s Catalan Paths with Odd Returns to the \(x\)-axis–Standard Version and Skew Catalan-Stanley Paths Open
For \(r=1,2,…, 6\), we obtain generating functions \(F^{(r)}_{k}(y)\) for words over the alphabet \([k]\), where \(y\) tracks the number of parts and \([y^n]\) is the total number of distinct adjacent \(r\)-tuples in words with \(n\) parts…
View article: Arndt-Carlitz compositions
Arndt-Carlitz compositions Open
Carlitz-compositions follow the restrictions of neighbouring parts $σ_{i-1}\neqσ_{i}$. The recently introduced Arndt-compositions have to satisfy $σ_{2i-1}>σ_{2i}$. The two concepts are combined to new and exciting objects that we call Arn…
View article: Generating functions and Abdelkader's random walk model
Generating functions and Abdelkader's random walk model Open
We link questions by Abdelkader about a class of random walks to \emph{Moran walks}.
View article: Motzkin paths of bounded height with two forbidden contiguous subwords of length two
Motzkin paths of bounded height with two forbidden contiguous subwords of length two Open
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 …
View article: Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection
Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection Open
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 alt…
View article: Peakless Motzkin paths of bounded height
Peakless Motzkin paths of bounded height Open
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 st…
View article: An online bin-packing problem with an underlying ternary structure
An online bin-packing problem with an underlying ternary structure Open
Following an orginal idea by Kn¨odel, an online bin-packing problem is considered where 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 terna…
View article: S-Motzkin paths with catastrophes and air pockets
S-Motzkin paths with catastrophes and air pockets Open
The so-called S-Motzkin paths are combined with the concepts 'catastrophes' and 'air pockets'.The enumeration is done by properly setting up bivariate generating functions which can be expanded using the kernel method.
View article: S-Motzkin paths with catastrophes and air pockets
S-Motzkin paths with catastrophes and air pockets Open
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.
View article: Skew Dyck paths without up–down–left
Skew Dyck paths without up–down–left Open
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.
View article: Deepest Nodes in Marked Ordered Trees
Deepest Nodes in Marked Ordered Trees Open
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 numb…
View article: Counting edges according to edge-type in $t$-ary trees
Counting edges according to edge-type in $t$-ary trees Open
Using the Lagrange inversion formula, $t$-ary trees are enumerated with respect to edge type (left, middle, right for ternary trees).
View article: Enumeration of partial Lukasiewicz paths
Enumeration of partial Lukasiewicz paths Open
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 …
View article: Partial Skew Motzkin Paths
Partial Skew Motzkin Paths Open
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 st…
View article: Skew Dyck paths without up--down--left
Skew Dyck paths without up--down--left Open
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.
View article: Skew Dyck Paths With Catastrophes
Skew Dyck Paths With Catastrophes Open
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.These two i…
View article: Partial Dyck paths with Air Pockets
Partial Dyck paths with Air Pockets Open
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 …
View article: Deepest nodes in marked ordered trees
Deepest nodes in marked ordered trees Open
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 num…
View article: Partial Sums of Horadam Sequences: Sum-free Representations via Generating Functions
Partial Sums of Horadam Sequences: Sum-free Representations via Generating Functions Open
Horadam sequences and their partial sums are computed via generating functions. The results are as simple as possible.