Subword counting and the incidence algebra

TL;DR

This paper generalizes the exponential relation between Pascal matrices and subword counting to incidence algebras of words and permutations, introducing restricted subword concepts inspired by permutation patterns.

Contribution

It extends the exponential identity to incidence algebras of words and permutations, defining restricted subwords and establishing a reciprocity theorem for pattern counting.

Findings

Generalized Pascal matrix exponential relation to incidence algebras

Introduced restricted subword concepts based on auxiliary index sets

Proved a reciprocity theorem linking subword counts with complementary restrictions

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 $R^c$.