Subword counting and the incidence algebra

Abstract: 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 $R^c$.