Séminaire Lotharingien de Combinatoire, 82B.67 (2019), 12 pp.

Ron M. Adin, Ira M. Gessel, Victor Reiner, and Yuval Roichman

Cyclic quasi-symmetric functions

Abstract. 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 s_λ 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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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