A new %26quot;dinv%26quot; arising from the two part case of the shuffle conjecture

作者：Duane A; Garsia A M; Zabrocki M^*

来源：Journal of Algebraic Combinatorics, 2013, 37(4): 683-715.

DOI：10.1007/s10801-012-0382-0

• Symmetric functions
• Macdonald polynomials
• Parking functions

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)(H) over tilde (mu) = F[B-mu](H) over tilde (mu). In a recent paper (Int. Math. Res. Not. 11:525-560, 2004), J. Haglund showed that the expression %26lt;Delta(hJ) E-n,E-k, e(n)%26gt; q, t-enumerates the parking functions whose diagonal word is in the shuffle 12... j boolean OR boolean OR 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). %26lt;br%26gt;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 C-p1 C-p2 ... C-pk 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 %26lt;Delta(hJ) C-p1 C-p2 ...C-pk 1, e(n)%26gt;, using this recursion to construct a new dinv statistic (which we denote ndinv), then showing that this polynomial enumerates the latter parking functions by t(area)q(ndinv).

• 出版日期2013-6