Polysymmetric functions, introduced by Asvin G and Andrew O'Desky as a generalization of symmetric functions, have natural connections to algebraic geometry and provide a foundation for further developments. In this paper, we study polysymmetric functions using stack partitions and develop combinatorial descriptions of several polysymmetric bases. We introduce two new signed polysymmetric bases and give explicit transition formulas among the monomial, homogeneous, elementary, power, and signed polysymmetric bases. These results extend many familiar identities from symmetric function theory to the polysymmetric setting.