PIMS-UVic Discrete Math Seminar: Amarpreet Rattan

For fixed n, consider the symmetric group S_n on the symbols 1,...,n and the set of *star* transpositions, the transpositions that contain the symbol n. A *star factorization* of a permutation b in S_n of length k is the writing of b as the product of k star transpositions. Goulden and Jackson (2009) showed that the number of such factorizations only depends on the conjugacy class of b and not on b itself, a remarkable fact given the special role the symbol n plays amongst star transpositions. We supply the first fully combinatorial proof of this fact that works for all lengths k, and our methods connect star factorizations to monotone factorizations. Star transpositions are connected to Jucys-Murphy elements, and we explain how our result can give expressions for the *transitive* image of certain symmetric functions evaluated at Jucys-Murphy elements.

This is joint work with Jesse Campion Loth (Heilbronn Institute and the University of Bristol).