Browsing by Author "Prodinger, Helmut"

Now showing 1 - 10 of 10
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    Application of Smirnov words to waiting time distributions of runs
    (Electronic Journal of Combinatorics, 2017) Freiberg, Uta; Heuberger, Clemens; Prodinger, Helmut
    Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The expectation and the variance of the waiting time for the first completed h-run of any letter (i.e., first occurrence of h subsequential equal letters) is computed. The expected waiting time for the completion of h-runs of j arbitrary distinct letters is also given.
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    Combinatorics : past and present
    (Stellenbosch : Stellenbosch University, 2006) Prodinger, Helmut
    Inaugural address delivered by Prof Helmut Prodinger on 3 May 2006, Stellenbosch University.
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    Contributions to the analysis of approximate counting
    (Stellenbosch : Stellenbosch University, 2016-03) Prodinger, Helmut; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (Mathematics)
    ENGLISH ABSTRACT : Approximate Counting is a classical technique with very challenging questions re- lated to its performance analysis. It is also somewhat similar to parameters around Digital Search trees. Surprising links to q-analysis and the theory of partitions exist. The author has contributed to the analysis during the last decades; the relevant pa- pers have been collected in this thesis. Some emphasis is on a recent development, namely, to introduce a parameter m (m counters instead of one).
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    Growing and destroying Catalan–Stanley trees
    (Discrete Mathematics and Theoretical Computer Science, 2018) Hack, Benjamin; Prodinger, Helmut
    Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan–Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan–Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan–Stanley tree of given size as well as for the “speed” of the growth process by comparing the size of a given tree to the size of its ancestors.
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    Protection number in plane trees
    (University of Belgrade - School of Electrical Engineering, 2017-10) Heuberger, Clemens; Prodinger, Helmut
    The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size n and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size n. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.
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    Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
    (De Gruyter, 2017) Prodinger, Helmut
    A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.
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    Some combinatorial matrices and their LU-decomposition
    (De Gruyter, 2020-02) Prodinger, Helmut
    Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.
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    Summations in Bernoulli's triangles via generating functions
    (University of Waterloo, 2017) Oliver, Kamilla; Prodinger, Helmut
    We revisit sums along straight lines of indices in Bernoulli triangles, and emphasize the use of generating functions as the appropriate tool. This leads to more direct and extended results, compared with a recent paper in this journal.
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    Visibility problems related to skip lists
    (Combinatorial Mathematics Society of Australasia, 2018) Prodinger, Helmut
    For sequences (words) of geometric random variables, visibility problems related to a sun positioned in the north-west are considered. This leads to a skew version of such words. Various parameters are analyzed, such as left-to-right maxima, descents and inversions.
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    A wide class of Combinatorial matrices related with Reciprocal Pascal and Super Catalan matrices
    (2019) Kilic, Emrah; Prodinger, Helmut
    Abstract. In this paper, we present a number of combinatorial matrices that are generalizations or variants of the super Catalan matrix and the reciprocal Pascal matrix. We present explicit formul for LUdecompositions of all the matrices and their inverses. Alternative derivations using hypergeometric functions are also given.