Motivated by the study of cluster algebras, in 2014 Adachi-Iyama-Reiten introduced the notion of $\tau$-tilting theory, primarily to address the deficiency of the classical tilting theory from the mutation point of view. This subject soon became the center of attention and found applications in various areas, including representation theory, homological algebra, lattice theory and combinatorics.

In this talk, we approach this flourishing concept from the combinatorial viewpoint. In particular, we show that a concrete description of morphisms between modules over certain algebras give an easy combinatorial tool that can be used to fully answer deep questions raised by the other approaches. We illustrate this by giving sufficient conditions for $\tau$-tilting infiniteness of some well-known algebras, so-called special biserial.

If time permits, we also show how this concept relates to the work of McConville on generalization of Tamari orders.