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.

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 a=(a_i)_i=1^∞ 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 μ^r_n(a) be the ratio between the maximum and minimum sizes of r consecutive pieces. Addressing a question of De Bruijn and Erdős, we define a family of sequences for which the asymptotic least upper bound of this ratio, μ_r(a) = lim sup_n→∞μ^r_n(a) , can easily be calculated. Hence, for small r, we present upper bounds on _r(a).

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.

A partial order < on [n] is naturally labelled (NL) if x < 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. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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.

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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.

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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.

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

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θ B ≈ 2:35526, and that it also contains every value at least θ B ≈ 2:35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ A ≈ 2:48187. Thus, we also refute his conjecture that the set of growth rates below λ A is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by Rényi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values.

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.