On a general class of non-squashing partitions

We define MM-sequence non-squashing partitions, which specialize to mm-ary partitions (studied by Andrews, Churchhouse, Erdös, Hirschhorn, Knuth, Mahler, Rødseth, Sellers, and 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 MM-sequence non-squashing partition functions, functions whose associated generating functions are non-modular. In particular, we obtain an exact formula for the mm-ary partition function, and by new methods, we recover Mahler’s and Erdös’ asymptotic for the mm-ary partition function. We also establish new results on factorial partitions, colored mm-ary partitions, and many other general families which have not been well understood or systematically studied. Finally, we conjecture Ramanujan-like congruences for the MM-sequence non-squashing partition functions.