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Brian Curtin

Publications and source records attributed to Brian Curtin.

6 recordsLinked to original sources
arXiv2026

Combinatorial structures connecting Latin squares and bireversible automata

This paper explores the theory of letter transducers, Mealy automata, and bireversible automata from a combinatorial perspective analogous to the theory of Latin squares. We view the sets of transitions of letter transducers as analogs of orthogonal arrays, and discuss two other combinatorial encodings of Mealy automata analogous to orthogonal pairs of Latin squares and to $(k,n)$-nets. We characterize various classes of automata (Mealy, reversible, invertible, bireversible) in terms of these combinatorial structures. In particular, we represent the inversion and dualization of transducers as parastrophisms. Further, similarly to the notion of the isotopisms of the quasigroups associated to Latin squares, we develop the notion of isotopisms of letter transducers generalizing transducer symmetry and preserving the class of bireversible automata.

cs.FL↗
arXiv2026

Generalizations of nets and Latin squares

We examine combinatorial structures that generalize $(k,n)$-nets, orthogonal arrays, and mutually orthogonal Latin squares. By a reticulation we mean a point set and two collections (types) of families of lines such that two lines of different types meet in exactly one point and each family of lines partitions the point set. The number of points incident with any line depends only upon the type of the line, and every point is incident with the same number of lines of a given type. Each choice of a single line family of each type leads to an arrangement of the points into a rectangular grid. Recording the line containing a given point in the corresponding position of an array gives a generalization of sets of mutually orthogonal Latin squares, dubbed a cooperative system. A cooperative system consists of a collection of column-Latin matrices and a collection of row-Latin matrices such that each column-Latin matrix is orthogonal to each row-Latin matrix. Recording lines which contain a given point as a tuple gives a generalization of certain orthogonal arrays, dubbed semi-orthogonal arrays. Notions of parastrophy and isotopy for cooperative systems, corresponding to permuting families of lines and lines within each family, are introduced. Constructions of small reticulations and three recursive constructions of larger reticulations are given.

math.CO↗
arXiv2025

Excluded power graphs of groups

Let $\mathcal{X}$ be a set of integers greater than one. The $\mathcal{X}$-excluded power graph of a group $G$ has vertex set $G$ and an edge from $g$ to each power of $g$ other than itself provided that the power is not divisible by any element of $\mathcal{X}$. When $G=H\times K$ for groups $H$ and $K$ with coprime orders, excluding the prime factors of $|H|$ yields a power graph with a quotient consisting of multiple copies of a quotient of the power graph (no exclusions) of $K$. Partial results for the semidirect product under the same conditions are given. We describe groups whose $\mathcal{X}$-excluded power graphs consist of disjoint directed cliques.

math.CO↗
arXiv2014

A group sum inequality and its application to power graphs

Let $G$ be a finite group of order $n$, and let $C_n$ be the cyclic group of order $n$. We show that $\sum_{g \in C_n} ϕ(\mathrm{o}(g))\geq \sum_{g \in G} ϕ(\mathrm{o}(g))$, with equality if and only if $G$ is isomorphic to $C_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of undirected edges in its directed power graph.

math.GR↗