  A Gauss diagram is a simple, combinatorial way to present a link. It is known
that any Vassiliev invariant may be obtained from a Gauss diagram formula that
involves counting subdiagrams of certain combinatorial types. In this paper we
present simple formulas for an infinite family of invariants in terms of
counting surfaces of a certain genus and number of boundary components in a
Gauss diagram associated with a closed braid. We then identify the resulting
invariants with partial derivatives of the HOMFLY-PT polynomial.
