Induction Relations in the Symmetric Groups and Jucys-Murphy Elements

Abstract

Transitive factorizations faithfully encode many interesting objects. The well-known ones include ramified coverings of the sphere and hypermaps. Enumeration of specific classes of such objects have been known for quite some time now. Hurwitz numbers, monotone Hurwitz numbers and hypermaps numbers were discovered using different techniques. Recently, Carrell and Goulden found a unified algebraic approach to count these objects in genus 0. Jucys-Murphy elements and centrality play important roles in establishing induction relations. Such a method is interesting in its own right. Its corresponding combinatorial decomposition is however intriguingly mysterious. Towards a understanding of direct combinatorial analysis of multiplication of arbitrary permutations, we consider methods, especially operators on symmetric functions, and related problems in symmetric groups.