Distance 2 Permutation Statistics and Symmetric Functions

This paper is presented as an undergraduate honors thesis by Christopher Severs under the supervision of Professor Jeffrey Remmel at UCSD. The aim of the project was to read material about permutation statistics and generating functions for the ring of symmetric functions and then address a problem not covered in the literature to date. In working on this project the author gained a much better understanding of this particular area of mathematics and some insight into how the process of research and writing works. The main goal of this paper is find generating functions for some new permutation statistics on certain subsets of the symmetric group Sn by defining an appropriate homomorphism of the ring of symmetric functions and then applying that homomorphism to a simple symmetric function identity. This idea was first introduced by Brenti [4]. In [4], Brenti introduces a homomorphism from the ring of symmetric functions to polynomials in a single variable that demonstrates a remarkable connection between permutation enumeration and symmetric functions. Specifically, if Λ is the ring of symmetric functions and ek is the kth elementary symmetric function, Brenti defines ξ : Λ → Q[x] by