On a general class of non-squashing partitions

Abstract

We define M-sequence non-squashing partitions, which specialize to m-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 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.