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 s_λ 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.

介绍了环状拟对称函数环及其非埃舍尔子环。自然基由基本的循环拟对称函数组成；对于非埃舍尔子环，它们作为复曲面P分区枚举器出现，对于具有总循环顺序的复曲面偏序集P。相关的结构常数由排列的循环洗牌决定。然后我们证明以下正性现象：对于每个非钩形 λ，Schur 函数s _λ的展开系数就基本循环拟对称函数而言是非负的。证明依赖于形状为 λ 的标准 Young tableaux (SYT) 上循环下降图的存在。该理论适用于循环 shuffle 和 SYT 通过循环下降的枚举。