Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods

Abstract: Many identities involving symmetric functions can be proved through bijective manipulations of tableaux. In this paper, we prove identities involving polysymmetric functions through bijections and sign-reversing involutions. In their paper titled "Polysymmetric functions and motivic measures of configuration spaces", Asvin G and Andrew O'Desky introduced the algebra of polysymmetric functions (PSym) which can be defined as the tensor product of copies of the symmetric functions algebra (Sym) where the $i$th tensor factor is scaled by $i$. On one hand, we can obtain bases of this algebra by taking tensor products of the bases of Sym. On the other hand, the Asvin G and Andrew O'Desky paper introduces non-pure tensor bases families $H$, $E$, $E<sup>+$, and $P$ that we call plethystic bases. In this paper, we present combinatorial interpretations of the entries of the transition matrices between all twelve pairs of distinct plethystic bases. We also provide new interpretations for six OEIS sequences that turn up in this context.