Cyclic quasi-symmetric functions

2018/11/13 by Adin, Ron M., Gessel, Ira M., Reiner, Victor +1 · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.05440

Abstract

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.

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• Inverse descent statistic for André and simsun permutations
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