## From parking functions to Gelfand pairs

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- by Kürşat Aker and Mahi̇r Bi̇len Can PDF
- Proc. Amer. Math. Soc.
**140**(2012), 1113-1124 Request permission

## 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 $\frac {1}{n+1}{ 2n \choose n }$, $n\geq 1$.## References

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## Additional Information

**Kürşat Aker**- Affiliation: Feza Gürsey Institute, Istanbul, Turkey
- Email: aker@gursey.gov.tr
**Mahi̇r Bi̇len Can**- Affiliation: Tulane University, New Orleans, Louisiana 70118
- Email: mcan@tulane.edu
- Received by editor(s): February 10, 2010
- Published electronically: November 16, 2011
- Communicated by: Jim Haglund
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**140**(2012), 1113-1124 - MSC (2010): Primary 20C30, 05A19, 05E18
- DOI: https://doi.org/10.1090/S0002-9939-2011-11010-4
- MathSciNet review: 2869097