Terry Visentin

A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus

The group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus. In particular, we show that there exists, for each genus, a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices.

The number of order-preserving maps of fences and crowns

We perform an exact enumeration of the order-preserving maps of fences (zig-zags) and crowns (cycles). From this we derive asymptotic results.

A Dual Plotkin Bound for

The effectiveness of quasi-Monte Carlo methods for numerical integration has led to the study of (T,M,S)-nets, which are uniformly distributed point sets in the Euclidean unit cube. A recent result, proved independently by Schmid/Mullen and Lawrence, establishes an equivalence between (T,M,S)-nets and ordered orthogonal arrays. In a paper of Martin and Stinson, a linear programming technique is described which gives lower bounds on the size of an ordered orthogonal array and, hence, on the quality parameter T of a (T,M,S)-net. In this correspondence, these ideas are used to derive a dual Plotkin bound for ordered orthogonal arrays. For a (T,M,S)-net in base b, this bound implies TgesM+1-S/1-bM-Slscr(lscr-1/b-1/b2 -middotmiddotmiddot-1/blscr), where lscr=1+lfloorM-T/Srfloor. The correspondence ends with an exploration of the implications of this bound relative to known tables and examples

(T,M,S)

We give the first example of a binary pattern which is Abelian 2-avoidable, but which contains no Abelian fourth power. We introduce a family

On Abelian 2-avoidable binary patterns

\{f_n\}_{n=1}^\infty

Factorisations for partition functions of random Hermitian matrix models

of binary morphisms which offer a common generalization of the Fibonacci morphism and the Abelian fourth-power-free morphism of Dekking. We show that the Fibonacci word begins with arbitrarily high Abelian powers, but for n ≥ 2, the fixed point of fn avoids xn+2 in the Abelian sense. The sets of patterns avoided in the Abelian sense by the fixed points of fn and fn+1 are mutually incomparable for n ≥ 2.

A Formulation for the Genus Series for Regular Maps

Abstract There is a remarkable relationship between the genus series for rooted maps and rooted quadrangulations that has been obtained by a character theoretic argument. Hitherto no combinatorial explanation for this has been given. In this paper we generalize this relationship to a larger set of maps for which such a relationship can be found and we ascertain some of the properties that a putative bijection must possess. Several examples of images of sets under this bijection are given. We give them in detail since they may contain useful information about the bijection.

A hypergeometric analysis of the genus series for a class of 2-cell embeddings in orientable surfaces

The partition functionZN, for Hermitian-complex matrix models can be expressed as an explicit integral over ℝN, whereN is a positive integer. Such an integral also occurs in connexion with random surfaces and models of two dimensional quantum gravity. We show thatZN can be expressed as the product of two partition functions, evaluated at translated arguments, for another model, giving an explicit connexion between the two models. We also give an alternative computation of the partition function for theφ4-model. The approach is an algebraic one and holds for the functions regarded as formal power series in the appropriate ring.