Tilings, Lie theory and combinatorics

Here we will review several approaches to words and one dimensional tilings, which produces combinatorics leading to a rationality theorem for some word invariant. We use the Kellendonk product to create an algebraic structure. Then we construct monoids (Kellendonk monoids) and monoid algebras (Kellendonk-Putnam algebras), which makes it possible to study the associated groups and Lie algebras. Also we obtain bialgebras and their standard modules. From these algebraic structures, we obtain interesting combinatorics, which controls the local indistinguishability of tilings. The combinatorics induce power series as invariants. We discuss Fibonacci sequences and their associated power series, which leads to several examples and new observations. Finally we discuss the rationality for the coefficients of the corresponding power series.