Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan-Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan-Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan-Lusztig polynomials of certain Weyl groups.