  We report about results revolving around Kostka-Foulkes and parabolic Kostka
polynomials and their connections with Representation Theory and Combinatorics.
It appears that the set of all parabolic Kostka polynomials forms a semigroup,
which we call {\it Liskova semigroup}. We show that polynomials frequently
appearing in Representation Theory and Combinatorics belong to the Liskova
semigroup. Among such polynomials we study rectangular $q$-Catalan numbers;
generalized exponents polynomials; principal specializations of the internal
product of Schur functions; generalized $q$-Gaussian polynomials; parabolic
Kostant partition function and its $q$-analog; certain generating functions on
the set of transportation matrices. In each case we apply rigged configurations
technique to obtain some interesting and new information about Kostka-Foulkes
and parabolic Kostka polynomials, Kostant partition function, MacMahon,
Gelfand-Tsetlin and Chan-Robbins polytopes. We describe certain connections
between generalized saturation and Fulton's conjectures and parabolic Kostka
polynomials; domino tableaux and rigged configurations. We study also some
properties of $l$-restricted generalized exponents and the stable behaviour of
certain Kostka-Foulkes polynomials.
