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R. A. Bailey

Publications and source records attributed to R. A. Bailey.

At least 19 recordsLinked to original sources
arXiv2026

Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs

An equitable partition of a graph $\Ga$ is a partition $\{V_1, \ldots, V_m\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a, b)$-perfect set in $\Ga$, where $a$ is the number of neighbours in $V_1$ of each vertex in $V_1$, and $b$ is the number of neighbours in $V_1$ of each vertex in $V_2$. In this paper we first derive general necessary conditions for a regular graph to admit two equitable partitions. As a corollary we obtain necessary conditions for the existence of an $(a,b)$-perfect set in a regular graph in terms of an arbitrary equitable partition. With the help of these results we then obtain necessary conditions for the existence of an $(a,b)$-perfect set in a normal Cayley graph in terms of the irreducible characters of the underlying group.

math.CO↗
arXiv2026

Efficient partially replicated block designs with each replication number one or two

We investigate block designs, under the A- and MV-criteria, when each treatment can have only one or two replications due to resource constraints, as can happen, for example, in early generation varietal trials. While these are commonly known as partially replicated designs, a key new feature of the present work is that no restriction about a constant block size is imposed on the subdesign consisting of the twice replicated treatments. This makes the derivation more challenging but allows us to entertain a wider class of competing designs and hence increases the flexibility of the results. Considering all treatments as equally important, design-independent, sharp lower bounds on the A- and MV-criteria are derived, so as to find highly efficient designs over this wider class. The roles of (a) linked block designs, (b) designs in an online catalog designtheory.org, and (c) partially balanced incomplete block designs, or duals thereof, as adapted to our setup, are explored at length. Illustrative examples are presented.

math.ST↗
arXiv2025

Permutation groups, partition lattices and block structures

Let $G$ be a transitive permutation group on $Ω$. The $G$-invariant partitions form a sublattice of the lattice of all partitions of $Ω$, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an \emph{orthogonal block structure}, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a \emph{poset block structure}, whose automorphism group is a \emph{generalised wreath product}. We examine permutation groups with these properties, which we call the \emph{OB property} and \emph{PB property} respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties. A famous theorem on permutation groups asserts that a transitive imprimitive group $G$ is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by~$G$). We extend this theorem to groups with the PB property, embeddng them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions. We have included background and historical material on these concepts.

math.GR↗
arXiv2025

The square array design

This paper is about the construction of augmented row-column designs for unreplicated trials. The method uses the representation of a $k \times t$ equireplicate incomplete-block design with $t$ treatments in $t$ blocks of size $k$, termed an auxiliary block design, as a $t \times t$ square array design with $k$ controls, where $k<t$. This can be regarded as an extension of the representation of a Youden square as a partial latin square for unreplicated trials. Properties of the designs, in particular in relation to connectedness and randomization, are explored. Particular attention is given to square array designs which minimize the average variances of the estimates of paired comparisons between test lines and controls and between test-line and test-line effects. The use of equireplicate cyclic designs as auxiliary block designs is highlighted. These provide a flexible and workable family of augmented row-column square array designs. Designs whose auxiliary block designs are not cyclic are also covered.

stat.ME↗
arXiv2025

The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem

It is known that if the derangements subgroup of a transitive non-regular permutation group is a proper subgroup, then it is a Frobenius--Wielandt kernel, and, conversely, minimal Frobenius--Wielandt kernels are proper derangements subgroups. We present here a short survey of the literature on this topic, and we show that, although there are no restrictions on the structure of the $p$-groups appearing as Frobenius--Wielandt complements, a $p$-group appears as a one-point stabiliser in a transitive non-regular permutation group with a proper derangements subgroup if and only if it satisfies a certain group-theoretic condition.

math.GR↗
arXiv2021

The geometry of diagonal groups

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over any group, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m>=3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions are equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to synchronization.

math.GR↗
arXiv2021

Diagonal groups and arcs over groups

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $Ω$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $m=2$), or generate a join-semilattice of dimension $m$ associated with a diagonal group over a base group $G$. In this paper we investigate what happens if we have $m+r$ partitions with $r\geq 2$, any $m$ of which are minimal elements of a Cartesian lattice. If $m=2$, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $m>2$, things are more restricted. Any $m+1$ of the partitions generate a join-semilattice admitting a diagonal group over a group $G$. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that $G$ must be abelian and must have three fixed-point-free automorphisms whose product is the identity. Under this hypothesis, such a structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order $p$, then the structure corresponds exactly to an arc of cardinality $m+r$ in the $(m-1)$-dimensional projective space over the field with $p$ elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order $q$ give examples where $G$ is the elementary abelian group of order $q$. These examples can be lifted to non-elementary abelian groups using $p$-adic techniques.

math.CO↗
arXiv2021

Uniform semi-Latin squares and their pairwise-variance aberrations

For integers $n>2$ and $k>0$, an $(n\times n)/k$ semi-Latin square is an $n\times n$ array of $k$-subsets (called blocks) of an $nk$-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. We show that when a uniform $(n\times n)/k$ semi-Latin square exists, the Schur optimal $(n\times n)/k$ semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J.P. Morgan for affine resolvable designs, and determine the uniform $(n\times n)/k$ semi-Latin squares with minimum PV aberration when there exist $n-1$ mutually orthogonal Latin squares (MOLS) of order $n$. These do not exist when $n=6$, and the smallest uniform semi-Latin squares in this case have size $(6\times 6)/10$. We present a complete classification of the uniform $(6\times 6)/10$ semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform $((n+1)\times (n+1))/((n-2)n)$ semi-Latin square when there exist $n-1$ MOLS of order $n$, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs (BIBDs) can be constructed from uniform semi-Latin squares. From the uniform $(6\times 6)/10$ semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 BIBDs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays $\mathrm{OA}(72,6,6,2)$.

math.ST↗
arXiv2021

The diagonal graph

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer $m$ and group $G$ (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set $Ω$, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph $Γ_D(G,m)$ is a Cayley graph for the group~$G^m$, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the Möbius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory.

math.CO↗
arXiv2020

Groups generated by derangements

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.

math.GR↗
arXiv2019

Substitutes for the non-existent square lattice designs for 36 varieties

Square lattice designs are often used in trials of new varieties of various agricultural crops. However, there are no square lattice designs for 36 varieties in blocks of size six for four or more replicates. Here we use three different approaches to construct designs for up to eight replicates. All the designs perform well in terms of giving a low average variance of variety contrasts. Supplementary materials are available online.

stat.ME↗
arXiv2018

Multi-part balanced incomplete-block designs

We consider designs for cancer trials which allow each medical centre to treat only a limited number of cancer types with only a limited number of drugs. We specify desirable properties of these designs, and prove some consequences. Then we give several different constructions. Finally we generalise this to three or more types of object, such as biomarkers.

stat.AP↗
arXiv2018

Sesqui-arrays, a generalisation of triple arrays

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a row and a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term \emph{sesqui-array} for such an array when only the condition on pairs of columns is deleted. Thus all triple arrays are sesqui-arrays. In this paper we give three constructions for sesqui-arrays. The first gives $(n+1)\times n^2$ arrays on $n(n+1)$ letters for $n\geq 2$. (Such an array for $n=2$ was found by Bagchi.) This construction uses Latin squares. The second uses the \emph{Sylvester graph}, a subgraph of the Hoffman--Singleton graph, to build a good block design for $36$ treatments in $42$ blocks of size~$6$, and then uses this in a $7\times 36$ sesqui-array for $42$ letters. We also give a construction for $K\times(K-1)(K-2)/2$ sesqui-arrays on $K(K-1)/2$ letters. This construction uses biplanes. It starts with a block of a biplane and produces an array which satisfies the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the \emph{Hussain chains} for the selected block contain no $4$-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of $3$-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.

math.CO↗
arXiv2014

Optimal cross-over designs for full interaction models

We consider repeated measurement designs when a residual or carry-over effect may be present in at most one later period. Since assuming an additive model may be unrealistic for some applications and leads to biased estimation of treatment effects, we consider a model with interactions between carry-over and direct treatment effects. When the aim of the experiment is to study the effects of a treatment used alone, we obtain universally optimal approximate designs. We also propose some efficient designs with a reduced number of subjects.

math.ST↗
arXiv2014

On optimality and construction of circular repeated-measurements designs

The aim of this paper is to characterize and construct universally optimal designs among the class of circular repeated-measurements designs when the parameters do not permit balance for carry-over effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz These results extend the work of Magda, Kunert, Filipiak and Markiewicz.

math.ST↗
arXiv2013

Randomization-based models for experiments: I. A chain of randomizations

We derive randomization-based models for experiments with a chain of randomizations. The estimation theory for these models leads to formulae for the estimators of treatment effects, their standard errors, and expected mean squares in the analysis of variance. We discuss the practicalities in fitting these models and outline the difficulties that can occur, many of which do not arise in two-tiered experiments.

math.ST↗