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

2012/05/28 by Adrian Duane, Duane, Adrian, Adriano M. Garsia +3
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1205.6128

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=(p₁,p₂,...,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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