  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
which 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 we prove within the article include
analogs to 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.
