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 τ ⊗ } , { p τ ⊗ } , and { m 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 σ ⊗ H δ , p σ ⊗ E δ , etc.