Transition matrices and Pieri-type rules for polysymmetric functions

Asvin G and Andrew O'Desky recently introduced the graded algebra PΛ 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 PΛ that are analogous to well-known classical formulas for Λ. In more detail, we consider pure tensor bases ⊗τ\, ⊗τ\, and ⊗τ\ for PΛ 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 δ\, δ\, ⁺δ\, and 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 E{g}ecio{g}lu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as s⊗σHδ, p⊗σEδ, etc.