A q-Analog of Foulkes' Conjecture

We propose a q-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials H-n (x; q) replaces the analogous difference of plethysms of complete homogeneous symmetric functions h(n)(x) in Foulkes' conjecture. At q = 0, we get back the original statement of Foulkes, and we show that our version holds at q = 1. We discuss further supporting evidence, as well as various generalizations, including a (q, t)-version.