Journal of Algebraic Combinatorics

, Volume 37, Issue 4, pp 683–715

A new “dinv” arising from the two part case of the shuffle conjecture

Article

DOI: 10.1007/s10801-012-0382-0

Cite this article as:
Duane, A., Garsia, A.M. & Zabrocki, M. J Algebr Comb (2013) 37: 683. doi:10.1007/s10801-012-0382-0
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Abstract

For a symmetric function F, the eigen-operator ΔF acts on the modified Macdonald basis of the ring of symmetric functions by \(\Delta_{F} \tilde{H}_{\mu}= F[B_{\mu}] \tilde{H}_{\mu}\). In a recent paper (Int. Math. Res. Not. 11:525–560, 2004), J. Haglund showed that the expression \(\langle\Delta_{h_{J}} E_{n,k}, e_{n}\rangle\)q,t-enumerates the parking functions whose diagonal word is in the shuffle 12⋯J∪∪J+1⋯J+n with k of the cars J+1,…,J+n in the main diagonal including car J+n in the cell (1,1) by tareaqdinv.

In view of some recent conjectures of Haglund–Morse–Zabrocki (Can. J. Math., doi:10.4153/CJM-2011-078-4, 2011), it is natural to conjecture that replacing En,k by the modified Hall–Littlewood functions \(\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1\) would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting the parking function touches the diagonal according to the composition p=(p1,p2,…,pk). We prove this conjecture by deriving a recursion for the polynomial \(\langle\Delta_{h_{J}} \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1 , e_{n}\rangle \), using this recursion to construct a new \(\operatorname{dinv}\) statistic (which we denote \(\operatorname{ndinv}\)), then showing that this polynomial enumerates the latter parking functions by \(t^{\operatorname{area}} q^{\operatorname{ndinv}}\).

Keywords

Symmetric functionsMacdonald polynomialsParking functions

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California, San DiegoLa JollaUSA
  2. 2.Mathematics and StatisticsYork UniversityTorontoCanada