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

Duane, A. ; Garsia, A.M. ; Zabrocki, M.

Journal of Algebraic Combinatorics, Tome 37 (2013) no. 4, pp. 683-715.

Résumé

Summary: For a symmetric function F, the eigen-operator $\Delta _{ F }$ 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 \dots J\cup \cup $J+$1 \dots $J+n with k of the cars J+1,$\cdots $,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},\cdots $,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 $t^{\operatorname{area}} q^{\operatorname{ndinv}}$ .

Détail

Keywords: symmetric functions, macdonald polynomials, parking functions