Statistics for special $q,t$-Kostka polynomials

Abstract:

Kirillov and Reshetikhin introduced rigged configurations as a new way to calculate the entries ${K_{\lambda \mu }}(t)$ of the Kostka matrix. Macdonald defined the two-parameter Kostka matrix whose entries ${K_{\lambda \mu }}(q,t)$ generalize ${K_{\lambda \mu }}(t)$. We use rigged configurations and a formula of Stembridge to provide a combinatorial interpretation of ${K_{\lambda \mu }}(q,t)$ in the case where $\mu$ is a partition with no more than two columns. In particular, we show that in this case, ${K_{\lambda \mu }}(q,t)$ has nonnegative coefficients.