, Volume 37, Issue 4, pp 683-715
Date: 26 Jun 2012

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

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Abstract

For a symmetric function F, the eigen-operator Δ F 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 t area q dinv.

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 E n,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=(p 1,p 2,…,p k ). 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 \(t^{\operatorname{area}} q^{\operatorname{ndinv}}\) .