On a general class of non-squashing partitions

We define M-sequence non-squashing partitions, which specialize to m-ary partitions Sloane, among others), factorial partitions, and numerous other general partition families of interest. We establish an exact formula, various combinatorial interpretations, as well as the asymptotic growth of M-sequence non-squashing partition functions, functions whose associated generating functions are non-modular. In particular, we obtain an exact formula for the m-ary partition function, and by new methods, we recover Mahler's and Erdös' asymptotic for the m-ary partition function. We also establish new results on factorial partitions, colored m-ary partitions, and many other general families which have not been well understood or systematically studied. Finally, we conjecture Ramanujan-like congruences for the M-sequence non-squashing partition functions.