HIGH ORDER CONGRUENCES FOR M -ARY PARTITIONS

. For a sequence M = ( m i ) ∞ i =0 of integers such that m 0 = 1 , m i ≥ 2 for i ≥ 1 , let p M ( n ) denote the number of partitions of n into parts of the form m 0 m 1 · · · m r . In this paper we show that for every positive integer n the following congruence is true:


Introduction
Given a positive integer n, a partition of n is an expression: where all the numbers n j are integers and 1 ≤ n 1 ≤ n 2 ≤ . . .≤ n k .Let p(n) denote the number of partitions of a natural number n.
One of the most classical problems in the theory of partitions is studying the arithmetic properties of certain sequences of partitions.It originates in the celebrated work of Ramanujan, who proved that p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), and p(11n + 6) ≡ 0 (mod 11) hold for each nonnegative integer n.Similar results were proved also for other types of partitions, that is, for functions counting only the partitions of a given number that satisfy certain constrains.
In this paper we focus on the case of m-ary partitions.More precisely, these are partitions such that every part is a power of a fixed number m.For example, all the 2-ary partitions of number 5 are 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 2 = 1 + 2 + 2 = 1 + 4.
In particular, the number of such partitions is equal to 4 =: b 2 (5).In general, there are no simple formulas for the number of m-ary partitions of n, and we will denote this number by b m (n).
In the context of m-ary partitions of various types, the problem of finding so called supercongruences, that is, congruences modulo high powers of m, is one of the most important parts of the theory.Churchhouse was the first to ask about arithmetic properties of the sequences of binary partitions.In [3], he proved that b 2 (n) is even if n ≥ 2.He also characterized numbers b 2 (n) modulo 4 and conjectured that for a fixed positive integer k we have for all positive integers n.This conjecture was proved by Rødseth in [8].
The above result was subsequently improved and extended to the case of m-ary partitions by Andrews [1], Gupta [5], and Rødseth and Sellers [9].Namely, they proved that where for a sequence (ε j ) r−1 j=1 ∈ {0, 1} r−1 we define Similar results were obtained for many types of m-ary partitions including the case of m-ary partitions with no gaps [2,6] and m-ary overpartitions [7].
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.More precisely, let M = (m i ) ∞ i=0 be a sequence of integers, m 0 = 1, m i ≥ 2 for i ≥ 1, and for r ≥ 0 let Then an M-ary partition of a positive integer n is an expression of the form for some nonnegative integers r 1 , . . ., r s .We denote the number of M-ary partitions of n by p M (n).
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].
where P j = p≤j p is the product of all primes up to j.Then for all integers n ≥ 1 and In the special case of partitions into factorials, the above conjecture can be stated as follows.
where σ is the same as in (2), c ∈ {1, 2}, and Unfortunately, the above conjectures are false in general.Indeed, we checked, using a Mathematica [10] that for example for n ∈ {2, 6, 8, 10, 12, 16}.In fact, Conjecture 1.2 failed in almost all cases we checked, that is, for sequences with r ≤ 5.However, we made an attempt to find and prove some weaker results of the same type.We focused on the situation in which σ = 0.Moreover, we prove the above conjectures only modulo a little worse modulus than the product For all positive integers a and b we define µ(a, b) := a (a,b) .Moreover, let for positive integers m and r.
The main result of the paper is the following.
Theorem 1.3.Let M = (m 0 , m 1 , . ..) be a (possibly finite) sequence of integers such that m 0 = 1 and m j ≥ 2 if j ≥ 1.Then for every positive integers n and r we have Let us provide some instances in which Conjecture 1.1 is true.In fact even stronger divisibility property holds if we assume that m r has only large prime factors for every r.

Auxiliary results
In our proof, we will use the main idea from [9].At first, we will connect the numbers of the form p M (m 1 . . .m r n − 1) with the numbers α m,r (i) defined in the following lemma.
Lemma 2.1.For any integers m ≥ 2 and n, r ≥ 1 there exist unique integers α m,r (i) such that Lemma 1] together with remarks after the proof.
In the proof, we will need the generating function of the sequence (p M (n)) ∞ n=0 .Let us denote it by F M (q): In particular, if For a positive integer m ≥ 2 let us define the following linear operator: The motivation behind considering the above operator is simple.The generating function of the sequence p M (m . We will explain this phenomenon in more details in the proof of Theorem 1.3 below.In the proof, we will need some properties of the operators U m that we gather in the next lemma.Lemma 2.2.For every m ≥ 2 the following properties hold. ( (2) If f (q), g(q) ∈ Z q , then Proof.This is a direct consequence of the definition of the operator U m .
As we explained earlier, the generating function of the sequence p M (m ∞ n=1 is an image of the series qF M (q) under the operator U mr • • • • • U m 1 .On the other hand, it will be possible to express the same sequence in a more manageable form.In order to do so, we introduce the following sequences of power series h r and H (m 1 ,...,mr) .Let for r ≥ 0.
Lemma 2.3.For each r and m we have where numbers α m,r (i) are the same as in Lemma 2.1.
Proof.Follows immediately from the definition of U m and Lemma 2.2.
For a sequence (m 1 , m 2 , . ..) of natural numbers greater than or equal to 2, let us define the following sequence of power series with integer coefficients (all these sequences depend on q so we skip the variable for simplicity of notation): and if H (m 1 ,...,m r−1 ) is defined for some r ≥ 2 then Example 2.4.For r ≤ 3 the series defined above satisfy the following equalities: (1) .
Proof.The number m will not change during the proof so we omit it and write α r (i) for α m,r (i) in order to simplify the notation.For every n we have: