SFU Discrete Math Seminar: Jesse Campion Loth

How many ways are there to write a given permutation as a product of m transpositions (i_1, j_1) (i_2, j_2) \dots (i_m, j_m)? For example, there are three ways to write (1,2,3) as a product of two transpositions: (1,2)(1,3), (1,3)(2,3) and (2,3)(1,2). This is an example of a permutation factorisation problem. There are a broad range of problems of this type, with links to algebraic geometry and graph embeddings. We will first give an overview of how problems of this type are usually approached using generating functions and character theory. We then show how a combinatorial approach can be used to give bijections between different classes of permutation factorisation. In particular, we will present the first fully combinatorial proof of the symmetry of star factorisations, answering a question of Goulden and Jackson [2009]. This is joint work with Amarpreet Rattan, who will give a seminar on a symmetric function approach to these problems in March.