Asvin G and Andrew O'Desky recently introduced the graded algebra P$Λ$ of polysymmetric functions as a generalization of the algebra $Λ$ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P$Λ$ that are analogous to well-known classical formulas for $Λ$. In more detail, we consider pure tensor bases $\{s^{\otimes}_τ\}$, $\{p^{\otimes}_τ\}$, and $\{m^{\otimes}_τ\}$ for P$Λ$ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for $Λ$. We find expansions in these bases of the non-pure bases $\{P_δ\}$, $\{H_δ\}$, $\{E^+_δ\}$, and $\{E_δ\}$ studied by Asvin G and O'Desky. The answers involve tableau-like structures generalizing semistandard tableaux, rim-hook tableaux, and the brick tabloids of Eğecioğlu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as $s^{\otimes}_σH_δ$, $p^{\otimes}_σE_δ$, etc.