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ISSN 1079-6762

 

$ q$-Eulerian polynomials: Excedance number and major index


Authors: John Shareshian and Michelle L. Wachs
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 33-45
MSC (2000): Primary 05A30, 05E05, 05E25
Published electronically: April 12, 2007
MathSciNet review: 2300004
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Abstract: In this research announcement we present a new $ q$-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our $ q$-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on $ q$-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker; (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev, and Lunts; (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville, and Vaughan and by Dollhopf, Goulden, and Greene; and (4) Stanley's chromatic symmetric functions.


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Additional Information

John Shareshian
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email: shareshi@math.wustl.edu

Michelle L. Wachs
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: wachs@math.miami.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-07-00172-2
PII: S 1079-6762(07)00172-2
Received by editor(s): October 16, 2006
Published electronically: April 12, 2007
Additional Notes: The first author was supported in part by NSF Grants DMS 0300483 and DMS 0604233, and the Mittag-Leffler Institute
The second author was supported in part by NSF Grants DMS 0302310 and DMS 0604562, and the Mittag-Leffler Institute
Communicated by: Sergei Fomin
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.