  Double Kostka polynomials are polynomials indexed by a pair of double
partitions. As in the ordinary case, double Kostka polynomials are defined in
terms of Schur functions and Hall-Littlewood functions associated to double
partitions. In this paper, we study combinatorial properties of those double
Kostka polynomials and Hall-Littlewood functions. In particular, we show that
the Lascoux-Schutzenberger type formula holds for double Kostka polynomials in
certain cases. Moreover, we show that the Hall bimodule introduced by
Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions
with two types of variables, which gives an alternate approach for their
result.
