Decomposition of Polysymmetric Functions and Stack Partitions

Abstract: Polysymmetric functions, introduced by Asvin G. and Andrew O'Desky, provide a new framework linking combinatorics and algebraic geometry through motivic measures in the Grothendieck ring of varieties. Building on this foundation and the combinatorial results of Khanna and Loehr, we study the non-tensor bases of the polysymmetric algebra, introducing a new basis ${H^+_{\tau}}$ and deriving explicit expansions among the families ${H_{\tau}}$, ${E_{\tau}}$, ${E^+_{\tau}}$, ${P_{\tau}}$, and ${H^+_{\tau}}$, where each basis is indexed by stack partitions (types) that generalize integer partitions by recording both degrees and multiplicities. Using the Pieri-type rule established by Khanna and Loehr for related families, we extend it to the $H^+$ basis, give a combinatorial description of the product of monomial polysymmetric functions, and identify an involution for the $P$ basis.