The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring 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. We then prove the following positivity phenomenon: 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 proof relies on the existence of a cyclic descent map on the standard Young tableaux (SYT) of shape λ. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents.
Bibliographical noteFunding Information:
VR was partially supported by NSF grant DMS-1601961. Acknowledgements
IMG was partially supported by grant no. 427060 from the Simons Foundation.
RMA and YR were partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no. 1970/18.
© 2021, The Hebrew University of Jerusalem.