## Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions.(English)Zbl 1087.05063

Summary: Let $$h_\lambda$$, $$e_\lambda$$, and $$m_\lambda$$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $$\lambda$$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $$m_\lambda$$ 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.

### MSC:

 05E05 Symmetric functions and generalizations 05A99 Enumerative combinatorics

### Keywords:

partition; permutations; Lyndon words
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