A combinatorial overview of the Hopf algebra of MacMahon symmetric functions

Abstract

A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we give a combinatorial overview of the Hopf algebra structure of the MacMahon symmetric functions relying on the construction of a Hopf algebra from any alphabet of neutral letters obtained in [18 G.-C. Rota and J. Stein, Plethystic Hopf algebras, Proc. Natl. Acad. Sci. USA 91 (1994) 13057–13061. 19. G.-C. Rota and J. Stein, Plethystic algebras and vector symmetric functions, Proc. Natl. Acad. Sci. USA 91 (1994) 13062–13066].