arXiv subjects
Publications and source records attributed to David Bevan.
After $n$ radial cuts of a circular cake, it is divided into $n$ slices. Call an adjacent pair of slices a portion. We exhibit an infinite sequence of cuts such that the ratio between the maximum and minimum sizes of a portion never exceeds 1.755. This improves on the trivial upper bound of 2, disproving a conjecture of Korsky.
Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and minimum sizes of $r$ consecutive pieces. Addressing a question of De Bruijn and Erd\H{o}s, we define a family of sequences for which the asymptotic least upper bound of this ratio, \[ \mu_r(\boldsymbol{a}) \;=\; \limsup_{n\to\infty}\mu^r_n(\boldsymbol{a}) , \] can easily be calculated. Hence, for small $r$, we present upper bounds on $\inf\mu_r(\boldsymbol{a})$.
As digital health solutions continue to reshape healthcare delivery, telehealth software applications have become vital for improving accessibility, continuity of care, and patient outcomes. This paper presents an analysis of designing a software application focused on Enhanced Telehealth Capabilities (ETHC) for palliative care, integrating across three socio-technical dimensions: quality, human values, and real-world. Designing for quality attributes -- such as performance, maintainability, safety, and security -- ensured that the system is technically robust and compliant with clinical standards. Designing for human values -- empathy, inclusivity, accessibility, and transparency -- helped enhance patient experience, trust, and ethical alignment. Designing for real-world -- through a multidisciplinary, experience-based co-design approach involving clinicians, patients, and carers that guided iterative cycles of prototyping, usability testing, and real-world evaluation -- ensured continuous refinement of features and alignment with clinical practice. The resulting telehealth software solution demonstrated that our socio-technical design framework was successful in producing a secure, equitable, and resilient digital health application. Our design approach can assist others designing software in health and other domains.
We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes.
We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the $M$-sum, which generalises the well-known notions of direct sum and skew sum, and enables a deeper understanding of the structure of permutations in these grid classes. Permutations which are indecomposable with respect to the $M$-sum play a crucial role in the structure of a grid class and of its subclasses, and this leads us to identify coils, a certain kind of permutation which corresponds to repeatedly traversing a chosen cycle in a particular manner. Harnessing this analysis, we give a precise characterisation for when a subclass of such a grid class is labelled well quasi-ordered, and we extend this to characterise (unlabelled) well quasi-ordering in certain cases. We prove that a large general family of these grid classes are finitely based, but we also exhibit other examples that are not, thereby disproving a conjecture from 2006 due to Huczynska and Vatter.
We exhibit a procedure to asymptotically enumerate monotone grid classes of permutations. This is then applied to compute the asymptotic number of permutations in any connected one-corner class. Our strategy consists of enumerating the gridded permutations, finding the asymptotic distribution of points between the cells in a typical large gridded permutation, and analysing in detail the ways in which a typical permutation can be gridded. We also determine the limit shape of any connected monotone grid class.
We explore how the asymptotic structure of a random permutation of $[n]$ with $m$ inversions evolves, as $m$ increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. The threshold for the appearance of a classical pattern depends on the greatest number of inversions in any of its sum indecomposable components.
A partial order $\prec$ on $[n]$ is naturally labelled (NL) if $x\prec y$ implies $x<y$. We establish a bijection between {3, 2+2}-free NL posets and 12-34-avoiding permutations, determine functional equations satisfied by their generating function, and use series analysis to investigate their asymptotic growth, presenting evidence of stretched exponential behaviour. We also exhibit bijections between 3-free NL posets and various other objects, and determine their generating function. The connection between our results and a hierarchy of combinatorial objects related to interval orders is described.
We explore how the asymptotic structure of a random $n$-term weak integer composition of $m$ evolves, as $m$ increases from zero. The primary focus is on establishing thresholds for the appearance and disappearance of substructures. These include the longest and shortest runs of zero terms or of nonzero terms, longest increasing runs, longest runs of equal terms, largest squares (runs of $k$ terms each equal to $k$), as well as a wide variety of other patterns. Of particular note is the dichotomy between the appearance and disappearance of exact consecutive patterns, with smaller patterns appearing before larger ones, whereas longer patterns disappear before shorter ones.
We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite number of scales.
We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${\boldsymbol\sigma}$ is a permutation chosen uniformly at random from the set of all permutations of $[n]$ with exactly $m=m(n)\ll n^2$ inversions. If $i<j$ are chosen uniformly at random from $[n]$, then ${\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)$ asymptotically almost surely. However, if $i$ and $j$ are chosen so that $j-i\ll m/n$, and $m \ll n^2/\log^2 n$, then $\lim_{n\to\infty}\mathbb{P}\big[{\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)\big]=\frac{1}{2}$. Moreover, if $k=k(n)\ll \sqrt{m/n}$, then the restriction of ${\boldsymbol\sigma}$ to a random $k$-point interval is asymptotically uniformly distributed over $\mathcal{S}_k$. Thus, knowledge of the local structure of ${\boldsymbol\sigma}$ reveals nothing about its global form. We establish that $\sqrt{m/n}$ is the threshold for local uniformity and $m/n$ the threshold for inversions, and determine the behaviour in the critical windows. As pointed out by a referee, there are flaws in the proofs that do not seem easily rectifiable (see comments on pages 9 and 15). So the results stated above have not been established.
An $n$-multiset of $[k]=\{1,2,\ldots, k\}$ consists of a set of $n$ elements from $[k]$ where each element can be repeated. We present the bivariate generating function for $n$-multisets of $[k]$ with no consecutive elements. For $n=k$, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern $DUD$, and we show how classical and novel statistics are transported by these bijections.
We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small domino subclass whose enumeration we relate to West-two-stack-sortable permutations and planar maps. The bounds are established by carefully combining copies of the dominoes in particular ways consistent with the structural characterisation. The lower bound depends on concentration results concerning the substructure of a typical domino, the determination of exactly when dominoes can be combined in the fewest distinct ways, and technical analysis of the resulting generating function.
A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations of m letters exist for every m \ge k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds.
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.
We determine the structure of permutations avoiding the patterns 4213 and 2143. Each such permutation consists of the skew sum of a sequence of plane trees, together with an increasing sequence of points above and an increasing sequence of points to its left. We use this characterisation to establish the generating function enumerating these permutations. We also investigate the properties of a typical large permutation in the class and prove that if a large permutation that avoids 4213 and 2143 is chosen uniformly at random, then it is more likely than not to avoid 2413 as well.
We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
A brief presentation of basic definitions and notation used in permutation patterns research.