Subwords and subword counting have long played an important role in
combinatorics on words.  For instance the Robinson-Schensted
correspondence between permutations and symmetric Young tableaux was
invented to calculate the length of the longest increasing subword of a
permutation.  Generalized binomial coefficients, which count subwords,
play a role in polynomial rings in non-commuting variables.

In automata theory, subword counting has also played a prominent role.  I.
Simon's work on piecewise testable languages in 1972 was the first
major result on regular languages involving subwords.   His results were
later interpreted by H. Straubing as showing that languages defined by
subword are precisely those recognized by unitriangular bit matrices.  S.
Eilenberg showed in 1976 that counting subwords modulo a prime p
corresponds to languages recognized by p-groups.  It is well known that
p-groups are precisely groups of unitriangular matrices over the p element
field.

In this talk we show how the combinatorics of multiplying block upper
triangular matrices over finite fields can be used to obtain a conceptual
proof, without case-by-case analysis, of results of Weil and Peladeau on
modular subword counting with regular constraints.  Some new results are
obtained in the process.  Links will be made with modular representation
theory of finite monoids if time permits.