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.

On the enumeration and congruences for m-ary partitions

“…The related works on such restricted m-ary partitions can be found in [2,4,11]. Moreover, in [5,7,10,15], a general class of non-squashing partitions was introduced and studied, which contains m-ary partitions as a special case.…”

“….} in the summation given by Folsom, Homma, Ryu and Tong [7,Theorem 1.5], it reduces to another j-fold summation expression for b m (n).…”

High order congruences for M-ary partitions

“…Folsom et al introduced in [4] the class of M-non-squashing partitions that generalizes ordinary m-ary partitions. Here, we call the M-non-squashing partitions the M-ary partitions for simplicity and due to their similarity with the ordinary m-ary partitions.…”

“…The natural question arises whether the congruences (1) can be generalized to a more general class of M-partitions, and if not, whether we can prove some weaker congruences. In fact, our main motivation is the following conjecture stated in [4].…”

Recurrence sequences connected with the m–ary partition function and their divisibility properties

“…For us, the two types of generalisations will be important. The first one, done by Folsom et al in [11] is the following. Let M = (m n ) ∞ n=0 be a fixed sequence of natural numbers such that m 0 = 1 and m j ≥ 2 for j ≥ 1.…”