I will discuss the surprising connections between finite-size scaling in bootstrap percolation models, limiting entropy rates for (Markov-type) stochastic processes, and the combinatorics of integer partitions. The limiting behavior of the percolation processes have an intrinsic combinatorial connection to certain probability sequences with gap conditions, which are further related to the generating functions of partitions with restricted sequence conditions. These generating functions are hypergeometric q-series that are of number-theoretic interest, as in some cases they are equal to the product of modular forms and Ramanujan's famous mock theta functions. In other cases, both the percolation processes and partition functions are best understood through entropy rate bounds for the probabilistic sequences, where techniques from the theory of linear operators are used to bound the dominant eigenvalues.