  A Gauss diagram is a simple, combinatorial way to present a knot. It is known
that any Vassiliev invariant may be obtained from a Gauss diagram formula that
involves counting (with signs and multiplicities) subdiagrams of certain
combinatorial types. These formulas generalize the calculation of a linking
number by counting signs of crossings in a link diagram. Until recently,
explicit formulas of this type were known only for few invariants of low
degrees. 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. We then identify the resulting
invariants with certain derivatives of the HOMFLYPT polynomial.
