  The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the
binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies
the beautiful relation $P=\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc.
on its superdiagonal and zeros elsewhere. We generalize this identity to the
incidence algebras $I(A^*)$ and $I(\mathcal{S})$ of functions on words and
permutations, respectively. In $I(A^*)$ the entries of $P$ and $H$ count
subwords; in $I(\mathcal{S})$ they count permutation patterns. Inspired by
vincular permutation patterns we define what it means for a subword to be
restricted by an auxiliary index set $R$; this definition subsumes both factors
and (scattered) subwords. We derive a theorem for words corresponding to the
Reciprocity Theorem for patterns in permutations: Up to sign, the coefficients
in the Mahler expansion of a function counting subwords restricted by the set
$R$ is given by a function counting subwords restricted by the complementary
set $R^c$.
