Cyclic quasi-symmetric functions

The ring of cyclic quasi-symmetric functions is introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; they arise as toric P-partition enumerators, for toric posets P with a total cyclic order. The associated structure constants are determined by cyclic shuffles of permutations. For every non-hook shape λ, the coefficients in the expansion of the Schur function sλ in terms of fundamental cyclic quasi-symmetric functions are nonnegative. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents.