## Abstract

In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q; t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended" Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.

Original language | English (US) |
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Pages | 39-50 |

Number of pages | 12 |

State | Published - Dec 1 2011 |

Event | 23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 - Reykjavik, Iceland Duration: Jun 13 2011 → Jun 17 2011 |

### Other

Other | 23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 |
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Country/Territory | Iceland |

City | Reykjavik |

Period | 6/13/11 → 6/17/11 |

## Keywords

- Affine permutations
- Catalan numbers
- Diagonal harmonics
- Ish arrangement
- Nabla operator
- Parking functions
- Shi arrangement

## ASJC Scopus subject areas

- Algebra and Number Theory