On the number of even roots of permutations

Let $\sigma$ be a permutation on $n$ letters. We say that a permutation $\tau$ is an even (resp. odd) $k$th root of $\sigma$ if $\tau^k=\sigma$ and $\tau$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).Comment: Second version, 11 page

A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

A ``hybrid method'', dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions--this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.Comment: 31 page

Sobre raíces k-ésimas en el grupo simétrico y en el grupo alternante

Sea G un grupo y k un entero positivo. Para un elemento g de G, decimos que h es una raíz k-ésima de g si se cumple que hk = g. Es un problema clásico determinar cuando un elemento de G tiene o no raíz k-ésima en G y en su caso calcular el número de dichas raíces (ver, por ejemplo, [7, 9, 10, 17, 18, 27, 28]). Uno de los grupos más estudiados en este sentido es el grupo simétrico que consiste de todas las biyecciones de un conjunto nito X de carnalidad n y la composición de funciones como operación binaria. El grupo simétrico se denota por Sn y a sus elementos se les conoce como permutaciones. Es conocido que las permutaciones se pueden clasificar en permutaciones pares y permutaciones impares. El conjunto de las permutaciones pares es un subgrupo del grupo simétrico al cual se le conoce como grupo alternante y se denota por An. En los artículos [4, 5, 6, 8, 20, 21, 24, 25, 33, 34] se pueden encontrar resultados relacionados con raíces en el grupo simétrico

Enumerating 0-simple semigroups

Computational semigroup theory involves the study and implementation of algorithms to compute with semigroups. Efficiency is of central concern and often follows from the insight of semigroup theoretic results. In turn, computational methods allow for analysis of semigroups which can provide intuition leading to theoretical breakthroughs. More efficient algorithms allow for more cases to be computed and increases the potential for insight. In this way, research into computational semigroup theory and abstract semigroup theory forms a feedback loop with each benefiting the other. In this thesis the primary focus will be on counting isomorphism classes of finite 0-simple semigroups. These semigroups are in some sense the building blocks of finite semigroups due to their correspondence with the Greens -classes of a semigroup. The theory of Rees 0-matrix semigroups links these semigroups to matrices with entries from 0-groups. Special consideration will be given to the enumeration of certain sub-cases, most prominently the case of congruence free semigroups. The author has implemented these enumeration techniques and applied them to count isomorphism classes of 0-simple semigroups and congruence free semigroups by order. Included in this thesis are tables of the number of 0-simple semigroups of orders less than or equal to 130, up to isomorphism. Also included are tables of the numbers of congruence free semigroups, up to isomorphism, with m Green’s ℒ-classes and n Green’s ℛ-classes for all mn less than or equal to 100, as well as for various other values of m,n. Furthermore a database of finite 0-simple semigroups has been created and we detail how this was done. The implementation of these enumeration methods and the database are publicly available as GAP code. In order to achieve these results pertaining to finite 0-simple semigroups we invoke the theory of group actions and prove novel combinatorial results. Most notably, we have deduced formulae for enumerating the number of binary matrices with distinct rows and columns up to row and column permutations. There are also two sections dedicated to covers of E-unitary inverse semigroups, and presentations of factorisable orthodox monoids, respectively. In the first, we explore the concept of a minimal E-unitary inverse cover, up to isomorphism, by defining various sensible orderings. We provide examples of Clifford semigroups showing that, in general, these orderings do not have a unique minimal element. Finally, we pose conjectures about the existence of unique minimal E-unitary inverse covers of Clifford semigroups, when considered up to an equivalence weaker than isomorphism. In the latter section, we generalise the theory of presentations of factorisable inverse monoids to the more general setting of factorisable orthodox monoids. These topics were explored early in the authors doctoral studies but ultimately in less depth than the research on 0-simple semigroups