We will explain a general combinatorial scheme which allows us to give an explicit action of a quantum affine or quantum toroidal algebra in a class of thin representations. Our main example for this talk will be the case of admissible $U_q \widehat{gl}(n)$ modules which are thin as evaluation modules over quantum toroidal algebra of type gl(n). The answer is given in terms of cylindrical plain partitions. This is a joint project with M. Bershtein, L. Rybnikov, and A. Trufanov.