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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

From parking functions to Gelfand pairs


Authors: Kürşat Aker and Mahir Bilen Can
Journal: Proc. Amer. Math. Soc. 140 (2012), 1113-1124
MSC (2010): Primary 20C30, 05A19, 05E18
Published electronically: November 16, 2011
MathSciNet review: 2869097
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Abstract: A pair $ (G,K)$ of a group and its subgroup is called a Gelfand pair if the induced trivial representation of $ K$ on $ G$ is multiplicity free. Let $ (a_j)$ be a sequence of positive integers of length $ n$, and let $ (b_i)$ be its non-decreasing rearrangement. The sequence $ (a_i)$ is called a parking function of length $ n$ if $ b_i \leq i$ for all $ i=1,\dots ,n$. In this paper we study certain Gelfand pairs in relation with parking functions. In particular, we find explicit descriptions of the decomposition of the associated induced trivial representations into irreducibles. We obtain and study a new $ q$-analogue of the Catalan numbers $ \frac {1}{n+1}{ 2n \choose n }$, $ n\geq 1$.


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Additional Information

Kürşat Aker
Affiliation: Feza Gürsey Institute, Istanbul, Turkey
Email: aker@gursey.gov.tr

Mahir Bilen Can
Affiliation: Tulane University, New Orleans, Louisiana 70118
Email: mcan@tulane.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11010-4
PII: S 0002-9939(2011)11010-4
Received by editor(s): February 10, 2010
Published electronically: November 16, 2011
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.