A new dinv arising from the two part case of the shuffle conjecture
For a symmetric function F, the eigen-operator ? 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. 11:525560, 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.
symmetric functions parking functions macdonald polynomials