Random sequential covering of a one-dimensional lattice by $k$-mers

In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts that increase coverage are accepted. The process continues indefinitely on an infinite substrate, and we analyze the dynamics of random sequential covering of $\mathbb{Z}$ using $k$-mers. We introduce a method that provides a comprehensive solution to the dynamics of this process. We derive explicit solutions for trimers, tetramers, and pentamers; we study numerically random sequential covering by longer polymers ($k>5$).