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In this paper, we will study the M\"obius polynomial, an invariant of rankedposets that arises in the study of splitting algebras. We will present aformula for the M\"obius polynomial of the direct product of posets in terms ofthe M\"obius polynomials of the factors. We will then use this formula tocalculate Hilbert series and graded trace generating functions associated tothe splitting algebras of the Boolean algebra and the poset of factors of anatural number n.

M\"obius Polynomials and Splitting Algebras of Direct Products of Posets