Permutation enumeration of the symmetric group and the combinatorics of symmetric functions
\textit{F. Brenti} [Pac. J. Math. 157, No. 1, 1-28 (1993; Zbl 0805.05089)] has shown that the polynomials obtained by enumerating conjugacy classes of the symmetric group \(S_n\) with respect to excedances are unimodal and symmetric, and thus arises naturally from the theory of symmetric functions. The authors give combinatorial proofs and interpretations for the various bases of symmetric functions under \(\xi\), a homomorphism (introduced by Brenti) on the ring of symmetric functions, where \(\xi(e_i)= (1- x)^{k- 1}/k!\) is the image of the elementary symmetric function \(e_k\), \(\xi(e_0)= 1\). Their combinatorial methods lead to numerous extensions, e.g. \(q\)-analogues which involve the inversion statistics, as well as a definition of new combinatorial objects---border rim hook tabloids. Numerous examples are included.
scientific article; zbMATH DE number 802348