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arXiv · 1205.6128

A new `dinv' arising from the two part case of the Shuffle Conjecture

Adrian Duane·Adriano M. Garsia·Mike Zabrocki
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Abstract

In a recent paper J. Haglund showed that a certain symmetric function expresion enumerates by t^{area} q^{dinv} of the parking functions whose diagonal word is in the shuffle of 12...j and 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). In view of some recent conjectures of Haglund-Morse-Zabrocki it is natural to conjecture that replacing E_{n,k} by the modified Hall-Littlewood functions would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j+1,...,j+n hits the diagonal according to the composition p=(p_1,p_2,...,p_k). We prove here this conjecture by deriving a recursion for the symmetric function expression then using this recursion to construct a new dinv statistic we will denote ndinv and show that this polynomial enumerates the latter parking functions by t^{area} q^{ndinv}.

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Adrian Duane, Adriano M. Garsia, Mike Zabrocki. 2012-05-28. A new `dinv' arising from the two part case of the Shuffle Conjecture. https://arxiv.org/abs/1205.6128

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