Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

Let hλ, eλ, and mλ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition λ respectively. We give combinatorial interpretations for the coefficients that arise in expanding mλ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.