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 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.
Original language | English |
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Pages (from-to) | 437-500 |
Number of pages | 64 |
Journal | Israel Journal of Mathematics |
Volume | 243 |
Issue number | 1 |
DOIs | |
State | Published - Jun 2021 |
Bibliographical note
Publisher Copyright:© 2021, The Hebrew University of Jerusalem.
Funding
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.
Funders | Funder number |
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MIT-Israel MISTI | |
National Science Foundation | DMS-1601961 |
Simons Foundation | |
Israel Science Foundation | 1970/18 |