From Tetris and TASEP to a one-dimensional lattice gas in a point-like external field

I will recall the application of geometric group theory methods to calculate the statistical sum of particle heaps in a box in $(1+1)$. It will be shown how a simple generalization of the considered method solves the Totally Asymmetric Simple Exclusion Process (TASEP) stationary state enumeration problem. Further we will see that the use of Vienneau's theorem allows to express statistical properties of the considered models through thermodynamic characteristics of one-dimensional lattice gas in a local (point) external field.