Papers by Arnold Knopfmacher
Discrete Mathematics Theoretical Computer Science Dmtcs, 2010
We investigate the probability that a random composition (ordered partition) of the positive inte... more We investigate the probability that a random composition (ordered partition) of the positive integer n has no parts occurring exactly j times, where j belongs to a specified finite 'forbidden set' A of multiplicities. This probability is also studied in the related case of samples Γ = (Γ1, Γ2, . . . , Γn) of independent, identically distributed random variables with a geometric distribution.
Aequationes mathematicae, Jun 21, 2024
Integer compositions of n are viewed as bargraphs with n circular nodes or square cells in which ... more Integer compositions of n are viewed as bargraphs with n circular nodes or square cells in which the ith part of the composition x i is given by the ith column of the bargraph with x i nodes or cells. The sun is at infinity in the north west of our two dimensional model and each node/cell may or may not be lit depending on whether it stands in the shadow cast by another node/cell to its left. We study the number of lit nodes in an integer composition of n and later we modify this to yield the number of lit square cells. We then count the number of columns being lit which leads naturally to those cases where only the first column is lit. We prove the theorem that the generating function for the latter is the same as the generating function for compositions in which the first part is strictly smallest. This theorem has interesting q-series identities as corollaries which allow us to deduce in a simple way the asymptotics for both the number of lit nodes and columns as n → ∞.
Journal of algebra combinatorics discrete structures and applications, Oct 6, 2023
In a Dyck path, a peak which is strictly (weakly) higher than all the preceding peaks is called a... more In a Dyck path, a peak which is strictly (weakly) higher than all the preceding peaks is called a strict (weak) left-to-right maximum. By dropping the restrictions for the path to end on the x-axis, one obtains Dyck prefixes. We obtain explicit generating functions for both weak and strict left-to-right maxima in Dyck prefixes. The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.
The art of discrete and applied mathematics, Dec 12, 2023
A subdiagonal composition of a positive integer is a composition with the property that the ith p... more A subdiagonal composition of a positive integer is a composition with the property that the ith part is less than or equal to i, and analogously a superdiagonal composition is a composition with the property that the ith part is greater than or equal to i. The generating functions for subdiagonal and superdiagonal compositions as well as for compositions with a larger class of lower and upper boundary conditions are obtained. The asymptotics estimates for the numbers in these various classes of compositions are found as the size n of the composition tends to infinity.
Contributions to Discrete Mathematics, May 17, 2022
We define the statistic of a push for words on an alphabet [k] and use this to obtain a generatin... more We define the statistic of a push for words on an alphabet [k] and use this to obtain a generating function measuring the degree to which an arbitrary word deviates from sorted order. Several subsidiary concepts are investigated: the number of cells that are not pushed, the number of already sorted columns, the number of cells that coincide before and after pushing, the fixed cells in words and finally, the frictionless pushes.
SIGSAM bulletin, Sep 1, 1997
Let Fs[Ar] denote the multiplicative semigroup of monic polynomials in one indeterminate X , over... more Let Fs[Ar] denote the multiplicative semigroup of monic polynomials in one indeterminate X , over a finite field Fg. We determine for each fixed q and fixed n the probability that a polynomial of degree n in Fg[X] has irreducible factors of distinct degrees only. These results are of relevance to various polynomial factorization algorithms.
Journal of Computational and Applied Mathematics, Feb 1, 1987
The connection between convergence of product integration rules and mean convergence of Lagrange ... more The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L, (1 <p < 00) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R. Our results apply to the weights exp(-x"/2), m = 2,4,6.. . , and for the Hermite weight (m = 2) extend results of Nevai [28] and Bonan [2] in at least one direction. The results are sharp in L,, , 1 < p G 2. As a consequence, we can improve results of Smith, Sloan and Opie [38] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. In the process, we prove a new Markov-Stieltjes inequality for Gauss quadrature sums, and solve a problem of Nevai on how to estimate certain quadrature sums.
arXiv (Cornell University), Dec 23, 1998
For words of length n, generated by independent geometric random variables, we consider the avera... more For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth left-to-right maximum, for fixed r and n → ∞.
Aequationes mathematicae
We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that... more We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that of a bargraph prefix where we relax the bargraph defining parameters and allow the bargraph to extend into negative territory. A bargraph prefix is any initial section of a bargraph. Generating functions for the prefixes which separately track the number of up, down and horizontal steps are found. The asymptotics for the average height of the last prefix step in the different bargraph extensions is given.
Quaestiones Mathematicae, Jan 14, 2022
Aequationes Mathematicae, Mar 5, 2021
We consider the bargraph representation of geometrically distributed words, which we use to defin... more We consider the bargraph representation of geometrically distributed words, which we use to define the water capacity of such words. We first find a bivariate capacity generating function for all geometrically distributed words, from which we compute the generating function for the mean capacity. Thereafter, by making extensive use of Rice’s method (Rice’s integrals) we derive an asymptotic formula for the average capacity of random words of length n as n tends to infinity.
Research Square (Research Square), Jun 21, 2023
We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that... more We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that of a bargraph prefix where we relax the bargraph defining parameters and allow the bargraph to extend into negative territory. A bargraph prefix is any initial section of a bargraph. Generating functions for the prefixes which separately track the number of up, down and horizontal steps are found. The asymptotics for the average height of the last prefix step in the different bargraph extensions is given.
Mathematika, Dec 1, 1988
A study is made of the length L(h, k) of the Euclidean algorithm for determining the g.c.d. of tw... more A study is made of the length L(h, k) of the Euclidean algorithm for determining the g.c.d. of two polynomials h, k in [X], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.
Journal of Computational and Applied Mathematics, Dec 1, 1989
An algorithm is considered, and shown to lead to various unusual and unique series expansions of ... more An algorithm is considered, and shown to lead to various unusual and unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also partially characterized.
Discrete Mathematics, 2001
For words of length n, generated by independent geometric random variables, we consider the avera... more For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth left-to-right maximum, for fixed r and n → ∞.
Дискретная математика, 2021
С помощью производящих функций получены оценки наличия (или отсутствия) разделяющих точек в разби... more С помощью производящих функций получены оценки наличия (или отсутствия) разделяющих точек в разбиениях и словах. Разделяющая точка - это такое значение $j$, что все значения слева от него не больше $j$, а все значения справа от него не меньше $j$. Получены также асимптотические формулы для чисел разбиений и слов без разделяющих точек, когда размер стремится к бесконечности. Они следуют из асимптотически эквивалентных верхних и нижних оценок, полученных с использованием вероятностных рассуждений.
Mathematica Slovaca
We study various combinatorial parameters in the set of inversion sequences of length n. First, w... more We study various combinatorial parameters in the set of inversion sequences of length n. First, we provide generating functions for 1-successions and then generalise this to p-successions. Thereafter we find the formulae for the total number of successions by extracting coefficients. We then do the same for runs of length r and the length of the ith run. Next, we derive a generating function for the number of inversions in inversion sequences themselves and show that inversion sequences with no inversions are counted by the Catalan numbers. Finally, viewing inversion sequences as bargraphs, we develop a generating function for their area.
Indian Journal of Pure and Applied Mathematics
The Ramanujan Journal, 2022
Diskretnaya Matematika, 2022
Композиции числа $n$ - это такие конечные последовательности положительных целых чисел $(\sigma_i... more Композиции числа $n$ - это такие конечные последовательности положительных целых чисел $(\sigma_i)_{i=1}^k$, что $$ \sigma_1+\sigma_2+\cdots +\sigma_k=n. $$ Композиция $n$ представляется в виде гистограммы площади $n$: высота $i$-го столбца гистограммы равна величине $i$-й части композиции. Мы рассматриваем клеточный периметр гистограммы, который равен числу граничащих с ней клеток. Получена производящая функция чисел гистограмм с заданным клеточным периметром. Средняя величина клеточного периметра вычисляется заново прямым перечислением. Наконец, найдено среднее значение клеточного периметра гистограммы с заданным полупериметром.
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Papers by Arnold Knopfmacher