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.