Arnold Knopfmacher
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Arnold Knopfmacher has a total of 63 co-authors. Here are the top co-authors by number of shared publications:
- Aubrey Blecher
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
29 shared publications - Margaret Archibald
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
21 shared publications - Charlotte Brennan
University of the Witwatersrand
19 shared publications John Knopfmacher
19 shared publications- Toufik Mansour
University of Haifa
17 shared publications Helmut Prodinger
9 shared publications
Latest Publications
- Open Access
- 24 July 2026
- Published by National Inquiry Services Center (NISC) in Quaestiones Mathematicae
- Vol. 49 (12) , 1845-1858
Peaks in superdiagonal bargraphs
A bargraph is a first quadrant, non-intersecting lattice path consisting of up, down and horizontal steps. Superdiagonal bargraphs are bargraphs in which all horizontal steps (except possibly for t...
- 24 July 2026
- Published by National Inquiry Services Center (NISC) in Quaestiones Mathematicae
- Vol. 49 (12) , 1845-1858
Open AccessPeaks in superdiagonal bargraphs
A bargraph is a first quadrant, non-intersecting lattice path consisting of up, down and horizontal steps. Superdiagonal bargraphs are bargraphs in which all horizontal steps (except possibly for t...
- Open Access
- 28 October 2025
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 20 (2) , 98-108
Dyck paths with first sojourn highest
A Dyck path is a lattice path in the first quadrant using steps $u=(1,1)$ and $d=(1,-1)$, starting at the origin and ending on the $x$-axis. A Dyck sojourn is a Dyck path with only one return to the $x$-axis. Various subclasses of Dyck paths are shown to be counted by Catalan numbers: Dyck paths with weakly highest first sojourn and semilength $n+1$ are counted by twice the $n$th Catalan number and Dyck paths with semilength $n+1$ where the second sojourn is weakly highest are counted by the $n$th Catalan number. Also Dyck paths in which the first sojourn is strictly highest are equinumerous to Dyck paths with exactly one peak of maximum height. Some of these equivalences are proved using bijections. Generating functions for all these subclasses are also developed in the paper. The asymptotics as $n\to\infty$ for the case where the first sojourn is strictly highest are explored.- 28 October 2025
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 20 (2) , 98-108
Open AccessDyck paths with first sojourn highest
A Dyck path is a lattice path in the first quadrant using steps $u=(1,1)$ and $d=(1,-1)$, starting at the origin and ending on the $x$-axis. A Dyck sojourn is a Dyck path with only one return to the $x$-axis. Various subclasses of Dyck paths are shown to be counted by Catalan numbers: Dyck paths with weakly highest first sojourn and semilength $n+1$ are counted by twice the $n$th Catalan number and Dyck paths with semilength $n+1$ where the second sojourn is weakly highest are counted by the $n$th Catalan number. Also Dyck paths in which the first sojourn is strictly highest are equinumerous to Dyck paths with exactly one peak of maximum height. Some of these equivalences are proved using bijections. Generating functions for all these subclasses are also developed in the paper. The asymptotics as $n\to\infty$ for the case where the first sojourn is strictly highest are explored. - Open Access
- 17 September 2025
- Published by Combinatorial Press in Online Journal of Analytic Combinatorics
- Vol. 2025 (20) , 1-19
Grid graphs of Catalan words
This paper fits in the intersection between two disparate areas of combinatorics. Namely, graph theory and the combinatorics of Catalan words. A Catalan word with n parts is defined as a word w = w1w2⋯wn over the set of positive integers in which w1 = 1 and 1 ≤ wk ≤ wk − 1 + 1 for k = 2, 3, …, n. In order to study the intersection of the two areas, a specific type of graph called a grid graph is defined for each Catalan word. The main thrust of the paper is investigating the degrees of vertices in grid graphs. For each of the possible fixed degrees i ∈ {1, 2, 3, 4}, we find generating functions DFi(x) where the coefficient of xn is the total number of vertices of degree i in all grid graphs with n parts.
- 17 September 2025
- Published by Combinatorial Press in Online Journal of Analytic Combinatorics
- Vol. 2025 (20) , 1-19
Open AccessGrid graphs of Catalan words
This paper fits in the intersection between two disparate areas of combinatorics. Namely, graph theory and the combinatorics of Catalan words. A Catalan word with n parts is defined as a word w = w1w2⋯wn over the set of positive integers in which w1 = 1 and 1 ≤ wk ≤ wk − 1 + 1 for k = 2, 3, …, n. In order to study the intersection of the two areas, a specific type of graph called a grid graph is defined for each Catalan word. The main thrust of the paper is investigating the degrees of vertices in grid graphs. For each of the possible fixed degrees i ∈ {1, 2, 3, 4}, we find generating functions DFi(x) where the coefficient of xn is the total number of vertices of degree i in all grid graphs with n parts.
- Open Access
- 7 April 2025
- Published by Springer Nature in Afrika Matematika
- Vol. 36 (2) , 1-17
Subdiagonal and superdiagonal partitions
In this paper we consider weakly monotonic sequences of positive integers $$\lambda _1,\lambda _2,\dots ,\lambda _k$$ such that either $$\lambda _i\le i$$ for all i (subdiagonal partitions) or $$\lambda _i\ge i$$ for all i (superdiagonal partitions). We provide generating functions for these partitions with respect to size and number of parts, using various approaches involving multivariate generating functions, functional equations, area statistics on lattice paths, and bijections.- 7 April 2025
- Published by Springer Nature in Afrika Matematika
- Vol. 36 (2) , 1-17
Open AccessSubdiagonal and superdiagonal partitions
In this paper we consider weakly monotonic sequences of positive integers $$\lambda _1,\lambda _2,\dots ,\lambda _k$$ such that either $$\lambda _i\le i$$ for all i (subdiagonal partitions) or $$\lambda _i\ge i$$ for all i (superdiagonal partitions). We provide generating functions for these partitions with respect to size and number of parts, using various approaches involving multivariate generating functions, functional equations, area statistics on lattice paths, and bijections. - Open Access
- 1 January 2025
- Published by National Library of Serbia in Applicable Analysis and Discrete Mathematics
- Vol. 19 (1) , 67-75
Water capacity of Dyck bridges
We define the notion of capacity, the ability to contain water, for Dyck bridges (also known as Grand-Dyck paths) of semi-length n. The starting point for the analysis is the generating function for Dyck paths according to area and number of steps. We obtain a generating function which tracks the capacity and semi-length of Dyck bridges. The total capacity of all Dyck bridges of semi-length n is investigated as well as the asymptotics thereof as n ? ?.
- 1 January 2025
- Published by National Library of Serbia in Applicable Analysis and Discrete Mathematics
- Vol. 19 (1) , 67-75
Open AccessWater capacity of Dyck bridges
We define the notion of capacity, the ability to contain water, for Dyck bridges (also known as Grand-Dyck paths) of semi-length n. The starting point for the analysis is the generating function for Dyck paths according to area and number of steps. We obtain a generating function which tracks the capacity and semi-length of Dyck bridges. The total capacity of all Dyck bridges of semi-length n is investigated as well as the asymptotics thereof as n ? ?.
- Open Access
- 28 November 2024
- Published by University of Primorska Press in The Art of Discrete and Applied Mathematics
- Vol. 7 (3) , #P3.02
Subdiagonal and superdiagonal compositions
The Art of Discrete and Applied Mathematics (ADAM) is a modern, dynamic, platinum open access, electronic journal that will publish high-quality articles of arbitrary length in contemporary discrete and applied mathematics in which neither the authors nor the readers incur any costs.
- 28 November 2024
- Published by University of Primorska Press in The Art of Discrete and Applied Mathematics
- Vol. 7 (3) , #P3.02
Open AccessSubdiagonal and superdiagonal compositions
The Art of Discrete and Applied Mathematics (ADAM) is a modern, dynamic, platinum open access, electronic journal that will publish high-quality articles of arbitrary length in contemporary discrete and applied mathematics in which neither the authors nor the readers incur any costs.
- Open Access
- 21 June 2024
- Published by Springer Nature in Aequationes mathematicae
- Vol. 99 (1) , 155-173
Casting light on integer compositions
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 $$n \rightarrow \infty $$ .- 21 June 2024
- Published by Springer Nature in Aequationes mathematicae
- Vol. 99 (1) , 155-173
Open AccessCasting light on integer compositions
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 $$n \rightarrow \infty $$ . - Open Access
- 31 December 2023
- Published by Combinatorial Press in Online Journal of Analytic Combinatorics
- Vol. Issue 18, (18) , 1-15
Shade in partitions
Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing) partitions of \( n \).- 31 December 2023
- Published by Combinatorial Press in Online Journal of Analytic Combinatorics
- Vol. Issue 18, (18) , 1-15
Open AccessShade in partitions
Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing) partitions of \( n \). - Open Access
- 14 October 2023
- Published by Springer Nature in Aequationes mathematicae
- Vol. 98 (4) , 1133-1149
Prefixes of bargraph paths
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.
- 14 October 2023
- Published by Springer Nature in Aequationes mathematicae
- Vol. 98 (4) , 1133-1149
Open AccessPrefixes of bargraph paths
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.