A class of \(q\)-symmetric functions arising from plethysm
This charming exposition introduces the reader to a new class of symmetric functions reliant on a parameter \(q\) and obtained via plethysm and classic symmetric functions. It quickly transpires that certain functions in this class have already appeared in the literature in relation to Jack polynomials, parking functions, and Ehrenborg's quasi-symmetric functions of non-crossing set partitions. However, not only are we introduced to this class but also to a variety of bases for it corresponding to classic bases of the symmetric functions, and combinatorial expansions for each basis in terms of classic symmetric functions, which in turn yield \(q\)-analogues of transition matrices between the classical bases of symmetric functions. Finally the author studies a map which acts on one new basis and he derives a \(q\)-analogue of Eulerian polynomials. This analogue has various properties determined before the paper closes with a variety of conjectures.
scientific article; zbMATH DE number 1525305
scientific article