Robert Brignall
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View article: Pin classes II: Small pin classes
Pin classes II: Small pin classes Open
Pin permutations play an important role in the structural study of permutation classes, most notably in relation to simple permutations and well-quasi-ordering, and in enumerative consequences arising from these. In this paper, we continue…
View article: On cycles in monotone grid classes of permutations
On cycles in monotone grid classes of permutations Open
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 …
View article: Mini-Workshop: Permutation Patterns
Mini-Workshop: Permutation Patterns Open
The study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments.…
View article: Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes
Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes Open
The juxtaposition of permutation classes $\mathcal{C}$ and $\mathcal{D}$ is the class of all permutations formed by concatenations $\sigma\tau$, such that $\sigma$ is order isomorphic to a permutation in $\mathcal{C}$, and $\tau$ to a perm…
View article: Labelled well-quasi-order in juxtapositions of permutation classes
Labelled well-quasi-order in juxtapositions of permutation classes Open
The juxtaposition of permutation classes $\mathcal{C}$ and $\mathcal{D}$ is the class of all permutations formed by concatenations $στ$, such that $σ$ is order isomorphic to a permutation in $\mathcal{C}$, and $τ$ to a permutation in $\mat…
View article: Uncountably many enumerations of well-quasi-ordered permutation classes
Uncountably many enumerations of well-quasi-ordered permutation classes Open
We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and i…
View article: Labelled well-quasi-order for permutation classes
Labelled well-quasi-order for permutation classes Open
While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi ord…
View article: A framework for minimal hereditary classes of graphs of unbounded clique-width
A framework for minimal hereditary classes of graphs of unbounded clique-width Open
We create a framework for hereditary graph classes $\mathcal{G}^δ$ built on a two-dimensional grid of vertices and edge sets defined by a triple $δ=\{α,β,γ\}$ of objects that define edges between consecutive columns, edges between non-cons…
View article: Uncountably Many Minimal Hereditary Classes of Graphs of Unbounded Clique-Width
Uncountably Many Minimal Hereditary Classes of Graphs of Unbounded Clique-Width Open
Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that $\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at most finitely many non…
View article: Uncountably many minimal hereditary classes of graphs of unbounded clique-width
Uncountably many minimal hereditary classes of graphs of unbounded clique-width Open
Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^α$, and show that $\mathcal{G}^α$ has unbounded clique-width unless $α$ contains at most finitely many non-zero letters. …
View article: Labeled well-quasi-order for permutation classes
Labeled well-quasi-order for permutation classes Open
While the theory of labeled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and using labeled well-quasi-order ar…
View article: Labelled well-quasi-order for permutation classes
Labelled well-quasi-order for permutation classes Open
While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi ord…
View article: Well‐quasi‐ordering and finite distinguishing number
Well‐quasi‐ordering and finite distinguishing number Open
Balogh, Bollobás and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove tha…
View article: Combinatorial Specifications for Juxtapositions of Permutation Classes
Combinatorial Specifications for Juxtapositions of Permutation Classes Open
We show that, given a suitable combinatorial specification for a permutation class $\mathcal{C}$, one can obtain a specification for the juxtaposition (on either side) of $\mathcal{C}$ with Av(21) or Av(12), and that if the enumeration for…
View article: Zeros of the Möbius function of permutations
Zeros of the Möbius function of permutations Open
We show that if a permutation $π$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $μ[π]$ of the interval $[1,π]$ is zero. As a consequence, we show that the proportion o…
View article: Intervals of permutations and the principal Möbius function
Intervals of permutations and the principal Möbius function Open
We show that the proportion of permutations of length $n$ with principal Möbius function equal to zero, $Z(n)$, is asymptotically bounded below by 0.3995. If a permutation $π$ contains two intervals of length 2, where one interval is an as…
View article: New bounds on the growth rate of 1324-avoiders
New bounds on the growth rate of 1324-avoiders Open
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 …
View article: A structural characterisation of Av(1324) and new bounds on its growth\n rate
A structural characterisation of Av(1324) and new bounds on its growth\n rate Open
We establish an improved lower bound of 10.271 for the exponential growth\nrate of the class of permutations avoiding the pattern 1324, and an improved\nupper bound of 13.5. These results depend on a new exact structural\ncharacterisation …
View article: Juxtaposing Catalan Permutation Classes with Monotone Ones
Juxtaposing Catalan Permutation Classes with Monotone Ones Open
This paper enumerates all juxtaposition classes of the form "$\mathrm{Av}(abc)$ next to $\mathrm{Av}(xy)$", where $abc$ is a permutation of length three and $xy$ is a permutation of length two. We use Dyck paths decorated by sequences of p…
View article: Characterising when the simple permutations of a class are monotone griddable
Characterising when the simple permutations of a class are monotone griddable Open
We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing…
View article: Characterising inflations of monotone grid classes of permutations
Characterising inflations of monotone grid classes of permutations Open
We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing…
View article: Juxtaposing Catalan permutation classes with monotone ones
Juxtaposing Catalan permutation classes with monotone ones Open
This paper enumerates all juxtaposition classes of the form "Av($abc$) next to Av($xy$)", where $abc$ is a permutation of length three and $xy$ is a permutation of length two. We use Dyck paths decorated by sequences of points to represent…
View article: 2 × 2 monotone grid classes are finitely based
2 × 2 monotone grid classes are finitely based Open
In this note, we prove that all 2×2 monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain 2×2 (generalized) grid cla…
View article: Linear Clique‐Width for Hereditary Classes of Cographs
Linear Clique‐Width for Hereditary Classes of Cographs Open
The class of cographs is known to have unbounded linear clique‐width. We prove that a hereditary class of cographs has bounded linear clique‐width if and only if it does not contain all quasi‐threshold graphs or their complements. The proo…
View article: $2\times 2$ monotone grid classes are finitely based
$2\times 2$ monotone grid classes are finitely based Open
In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (gen…