We introduce a class of f(t)-factorials, or f(t)-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind that arise as the coefficients of the symbolic polynomial expansions of these f-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles include analogs of known expansions of the ordinary Stirling numbers by p-order harmonic number sequences, through the definition of a corresponding class of p-order f-harmonic numbers.