Symmetric Unimodality Index

A conformal partition function Pm n (s), which arised in the theory of Diophantine equations supplemented with additional restrictions, is concerned also with the reciprocal and skew-reciprocal algebraic equations based on the polynomial invariants of the symmetric group Sn. Making use of the relationship between Gaussian generating function for conformal partitions and Molien generating function for usual restricted partitions we derived the analytic expressions for Pm n (s). The unimodality indices for the reciprocal and skew-reciprocal equations were found. An existence of algebraic functions λn(xi) which are invariant upon the action both the finite group G ⊂ Sn, permuting n positive variables xi, and conformal group W, inverting both the function λn and the variables xi, is discussed. Pacs: Number theory, Invariant theory