MACMAHON’S PARTITION ANALYSIS XIII: SCHMIDT TYPE PARTITIONS AND MODULAR FORMS

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p(n), the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. AMS Mathematics Subject Classification. Primary: 05A17, 11P83, 11F20; secondary: 05A15, 11F03, 11Y99.

The point of this paper is the observation that this theorem has a very natural setting in MacMahon's Partition Analysis. The advantage of this approach is that it leads to a variety of related theorems of which the following is the most immediate.
The study of MacMahon's Partition Analysis has been the topic of twelve papers we wrote jointly on this topic. Partition Analysis is ideal to study questions of this nature. One of our hopes in launching our study is that we might find new classes of partitions whose generating functions are modular forms. In the following sections, a number of Schmidt type partitions arising from partitions on various graphs will be seen to have modular forms as generating functions. For example, in [2], we considered "plane partition diamonds"; i.e., partitions whose parts, a i , lie on the following graph, Figure 1, with each directed edge indicating ≥.  Now, taking such plane partition diamonds of unrestricted length (i.e., n → ∞), and if instead of adding up all the parts, we only add a 1 + a 4 + a 7 + · · · + a 3k+1 , we will find in Theorem 4 that the generating function is After giving a short review of Partition Analysis in Section 2, in Section 3 we prove Schmidt's original result, Theorem 1, and also Theorem 2. Moreover, we extend Theorem 2 to partitions with three colors; see Theorem 3 The remainder of our article is devoted to other appealing applications of Schmidt's original idea. In Section 4 we sum the links of partition diamonds and prove Theorem 4. In Section 5 we turn to the aspect of modular functions and related arithmetic properties of the coefficients of the constructed counting functions for Schmidt type partitions. Our primary tool in these investigations is Smoot's implementation [14], the Mathematica package RaduRK, of Radu's Ramanujan-Kolberg algorithm [12]. With this package we prove a variety of identities which, as immediate corollaries, imply divisibility properties. To supplement the "computer proof" of Theorem 5 given in Section 5, in Section 6 we present a classical proof which invokes two identities from the work of Nathan Fine [5]. In Section 7 we sum the links of k-elongated partition diamonds, objects introduced in [4]. In (7.3) we define the corresponding infinite family of generating functions, (D k (q)) k≥1 , counting Schmidt type partitions, and prove a variety of q-series relations and congruences for k = 2 and k = 3. In Subsection 7.1, some conjectures concerning congruences living on arithmetic subsequences of Schmidt type partition numbers for k = 2 are stated.

Partition Analysis: Basic Facts
We shall treat Theorem 1 in great detail to make clear how Partition Analysis is the ideal tool for managing partition questions of this nature. To this end, we prepare by providing some basic facts of MacMahon's method.
The MacMahon operator Ω ≥ is given by where the domain of the A s1,...,sr is the field of rational functions over C in several complex variables and the λ i are restricted to a neighborhood of the circle |λ i | = 1. In addition, the A s1,...,sr are required to be such that any of the series involved is absolute convergent within the domain of the definition of A s1,...,sr .
The only application we need for Theorem 1 is where r, s ∈ Z ≥0 . This result follows easily, .
To connect to Partition Analysis, we first consider the generating function of Schmidt partitions of n having exactly one summand a 1 . It is given by the coefficient of q n in .
The full generating function in x 1 and x 2 translates into where the last equality is by (2.2). Consequently, the counting function for the number of Schmidt partitions having exactly one summand is To determine the closed form of Next, we ask the program to eliminate all the slack variables λ i , The program, in this step, applies the elimination rule (2.2) to eliminate successively λ 1 , λ 2 , and λ 3 . Consequently, 1 The package is freely available at https://combinatorics.risc.jku.at/software upon password request via email to the second named author.
Summarizing, the experiments with the Omega package suggest that is the counting function for the number of Schmidt partitions a 1 + a 3 + · · · + a 2k−1 formed by exactly k summands. Finally, recall This would prove Theorem 1, once the form (2.6) for Schmidt partitions into k parts is proven. This is done in the next section, again by using Partition Analysis.

Proof of Theorem 1 and Partitions with two and three colors
For the proof of Theorem 1 and of similar results the following Lemma is crucial.
Lemma 3.1. For the n-variable generating function for partitions with n parts ≥ s ∈ Z ≥0 and difference at least r ∈ Z ≥0 between parts, and where x i keeps track of the ith part of the partition, Proof. We rewrite the left-hand side of (3.1) using MacMahon's operator Ω ≥ , and then successively apply (2.2), Finally, another application of (2.2), this time with B = 0, proves (3.1).
Corollary 1. The counting function for Schmidt partitions a 1 + a 3 + · · · + a 2k−1 formed by exactly k summands is of the form as in (2.6).
We conclude this Section with another application of Lemma 3.1.
Theorem 3. Let u(n, k) denote the number of partitions a 1 + a 2 + · · · + a 3k satisfying a 1 > a 2 > · · · > a 3k ≥ 0 and n = a 1 + a 4 + a 7 + · · · + a 3k−2 . Let v(n, k) denote the number of three-color partitions of n using exactly k parts of the first color with minimal difference 2 between parts, exactly k parts of the second color with minimal part-difference 1, and maximally k parts of the third color. Then Proof. By Lemma 3.1 with r = 1, s = 0 and n = 3k,

Summing Links of Partition Diamonds
In the Introduction we mentioned plane partition diamonds a 1 +a 2 +· · ·+a 3n+1 with a 1 ≥ a 2 ≥ · · · ≥ a 3n+1 ≥ 0 of length n; see Figure 1. Now, for such plane partition diamonds of unrestricted length (i.e., n → ∞) our Schmidt type condition consists of adding up the parts on the nodes linking the diamonds, a 1 + a 4 + a 7 + · · · + a 3k+1 .
For example, there are 13 such plane partition diamonds that yield 2: four with source 1 as shown in Figure 2, and the nine with source 2, Figure 3. Theorem 4. The generating function for the Schmidt type partitions obtained by adding the summands a 1 + a 4 + · · · + a 3k+1 , k ≥ 0, at the linking nodes in the plane partition diamonds of unrestricted length is given by Remark. There is a variety of combinatorial interpretations of the infinite product D(q). For example, it generates four-color partitions in which one of the colors has distinct parts.
Proof. In [2, Thm. 2.1] we proved that the generating function for the plane partition diamonds of length n, where x i keeps track of the part on the ith node (as indicated in Fig. 1), is given by To obtain those partitions where we consider we set x 3i+1 = q, and x j = 1 if j ≡ 1 (mod 3). As a result, X 3i+1 = X 3i+2 = X 3i+3 = q i+1 , i ≥ 0, and (4.2) becomes and letting n → ∞, we obtain the product (4.1).

Corollary 2.
Let d(m) be the number of partitions considered in Theorem 4. Then for m ≥ 0, In the next section, this corollary will be put into a broader context involving modular functions.

Witness Identities for Partition Diamond Congruences
As pointed out in the Introduction, one of the goals of our study of this type of partitions, enabled by the usage of Partition Analysis, is to give combinatorial constructions of modular forms which, in turn, will give rise to new arithmetical theorems. In this section will illustrate this aspect with the modular form D(q) introduced in Theorem 4.
Our first result is the first statement of Corollary 2 in the form of a witness identity.
Theorem 5. Let d(m) be the number of partitions considered in Theorem 4. Then Today, identities like (5.1) can be proven with computer algebra programs which implement Radu's Ramanujan-Kolberg algorithm [12]. We will apply the Mathematica package RaduRK by Nicolas Smoot [14] which is very convenient to use. 2 To prepare for its usage, follow the installation instructions given in [14], and invoke it within a Mathematica session as follows: Before running the program, one needs to set the two global key variables q and t: Proof. The algorithmic proof of (5.1) is done with the procedure call After a few seconds, Smoot's package delivers the proof in the form, The interpretation of the output is as follows: In the output expression P m,r (j) the abbreviation r := (r δ ) δ|M is used; i.e., here r = (−4, 1). 2 The package is freely available at https://combinatorics.risc.jku.at/software upon password request via email to the second named author.
• The last two entries in the procedure call RK[4,2,{−4,1},2,1] correspond to the assignment m = 2 and j = 1, which means that we are interested in the generating function In the output expression P m,r (j) these parameters m and j are used; i.e., here P m,r (j) = P 2,r (1) with r = (−4, 1).
• The output P m,r (j) = P 2,(−4,1) (1) = {1} means that there exists an infinite product • The output presents a solution to the following task: find a modular function t ∈ M (Γ 0 (N )) and polynomials p g (t) such that In general, the elements of the finite set AB constitute a C[t]-module basis of M (Γ 0 (N )), resp. of a large subspace of M (Γ 0 (N )). The elements g of AB are C-linear combinations of modular functions in M (Γ 0 (N )) which are representable in infinite product form such as f 1 (q) and t. In the specific case under consideration, the program delivers (5.2), which means, This is (5.1).
Remark. For the definition of notions such as Γ 0 (N ) or M (Γ 0 (N )), together with a general introduction to Radu's Ramanujan-Kolberg algorithm, see [11]. For the correctness proof and details of the algorithm, resp. of the implementation, see [12], resp. [14].
The next result is a witness identity which implies the second statement of Corollary 2.
Theorem 6. Let d(m) be the number of partitions considered in Theorem 4. Then To see that this indeed implies the second statement of Corollary 2, one uses (1−x) 4 ≡ (1−x 2 ) 2 (mod 4) repeatedly to obtain, Proof. For the algorithmic proof of (5. which, after a few seconds, computes the constituents of the witness identity (5.4) in the form, Remark. Notice that this time N = 8; i.e., the witness identity of the form, The next result is a Ramanujan-Kolberg relation which, as a witness identity, is a q-series refinement of the fact that 4 | a(2m + 1). the program produces the Ramanujan-Kolberg type identity (5.5) as follows:

Theorem 7. Let d(m) be the number of partitions considered in Theorem 4. Then
Remark. Again the relation involves modular functions in M (Γ 0 (N )) with N = 8. But now, according to the output P m,r (j) = {1, 3}, the witness identity involves a product of generating functions, with the polynomial p 1 (t) as given in the output Out [9]. Identities involving products in this form were first studied in systematic manner by Kolberg [8]. The entry "Common Factor" in the output refers to the common factor 16 of all the coefficients of p 1 (t).

A Classical Proof of Theorem 5
Let d(m) be the number of partitions considered in Theorem 4. In Section 5 we presented a "computer-proof" of the infinite product representation of m≥0 d(2m+1)q m stated in Theorem 5. We find it instructive to show how this identity, (5.1), can be proven with classical means. To this end, recall To simplify further, we recall two identities arising as special cases of a more general framework studied by Nathan Fine. The first one is [5, p. 76, (31.51)], where σ is the sum of divisors function. 3 The second identity is [5, p. 76, (31.54)], Now, by "summing the even part", (by (6.2)). Consequently, This completes the proof of (5.1).

Summing Links of k-elongated Partition Diamonds
In this section we will continue the theme of summing links of partition diamonds. To this end, we consider k-elongated plane partition diamonds introduced in [4]. The case k = 1 corresponds to using square-shaped building blocks of plane partition diamonds as depicted in Figure 1.
Instead of glueing squares together as in Figure 1, we take as building blocks k-elongated partition diamonds, which are configurations as shown in Fig. 4.

≥0
: the a i satisfy the order relations in Figure 5}.
As a consequence, the generating function of partition numbers produced by summing the links of k-elongated plane partition diamonds of length n ( Figure 5) is obtained by the substitutions For n, k ∈ Z ≥1 we define .
The substitution as in (7.1) gives Consequently, (7.2) implies D n,k (q) = 1 (q; q) 2k+1 n · 1 1 − q n+1 · (q 2 ; q 2 ) k n (q; q) k n , which in the limit n → ∞ turns into Notice that for k = 1 we have The remaining part of this section presents various observations on D k (q), respectively on the partition numbers d k (m), for k = 2 and k = 3.
7.1. Arithmetic properties of d 2 (m). This subsection is devoted to a study of arithmetic properties of the coefficients of the generating function for 2-elongated plane partitions, In this case the building blocks, 2-elongated diamonds of length 1, are of hexagonal shape. Divisibility by powers of 3 seem to be among the most striking properties of the d 2 (m). We begin with a witness identity.
Another corollary of Theorem 8 concerns the general distribution of even and odd partitions numbers d 2 (3m + 2). Corollary 6. Let d 2 (m) be the number of partitions as in (7.4). Then Proof. Notice that 33 is the only odd coefficient occuring in the polynomial on the right-hand side of (7.5), which implies, The last equivalence is by (1 − x) 2 ≡ 1 − x 2 (mod 2).
In a similar fashion, relation (7.5) implies further identities of the form (7.6), for example, with respect to mod 4 and mod 8. This principle applies in general: identities as (7.5) often give rise to equivalences such as (7.6), provided the coefficients of the polynomial in t show sufficiently "nice" patterns when taken modulo suitable integers. To illustrate this aspect in the given context, we restrict to showing only a small sample of such results.
Corollary 7. Let d 2 (m) be the number of partitions as in (7.4). Then The procedure calls RK[6, 2, {−7, 2}, 9,5] and RK[6, 2, {−7, 2}, 9,8] deliver relations with polynomials in t of degree 14, which imply the following divisibilities.   Applying the RaduRK package, based on computer algebra, to sequences d 2 (3 k m + j) for k ≥ 4 runs up against computational limits when using a standard laptop. Hence we conclude this subsection with several conjectures made on the basis of numerical computations. Owing to lack of numerical evidence the following conjecture concerning an infinite Ramanujan type family of divisibilities is more daring.
Conjecture 3. Let d 2 (m) be the number of partitions as in (7.4). Then for m ≥ 0, where the integers j k are chosen such that 8j k ≡ 1 (mod 3 k ) and 1 < j k < 3 k . The first j k are 2, 8, 17, 71, 152, etc.
Note added in proof. Ralf Hemmecke, using his implementation of Radu's Ramanujan-Kolberg algorithm from the QEta package [6], succeeded to produce a computer proof of Conjecture 1 for the case j = 71. In addition, Nicolas Smoot succeeded to prove the instance j = 152 of Conjecture 2. Moreover, Smoot was able to find a refinement of Conjecture 3 together with a proof.
7.2. Arithmetic properties of d 3 (m). This subsection is devoted to a study of arithmetic properties of the coefficients of the generating function for 3-elongated plane partitions In this case the building blocks, 3-elongated diamonds of length 1, are of octagonal shape. We present a couple of a witness identities which again imply various congruences.
Corollary 11. Let d 3 (m) be the number of partitions as in (7.20). Then Proof. Notice that the only odd coefficient occuring in the polynomial on the right-hand side of (7.21) is 1, the coefficient of t 6 . This implies, Remark. It is interesting to note that the right-hand side of (7.23) leads us back to our first result on Schmidt type partitions, Theorem 2. Moreover, as already remarked, the structure of the polynomial t(80 + 40t + t 2 ) gives rise to further equivalences, e.g., modulo 5 and 8.
Theorem 11. Let d 3 (m) be the number of partitions as in (7.20).

Conclusion
This paper hopefully will spur efforts to find further natural arithmetic/combinatorial objects generated by modular forms. Applications most often arise from the combinatorial side with subsequent important information being supplied by the fact that the generating functions are modular forms. The richness of results found from these few instances considered here suggests that much awaits.
Concerning the topical area of this paper, the Conjectures 1, 2, and 3 stated in Subsection 7.1 seem to be particularly challenging, especially the infinite family (7.19) of Ramanujan type congruences. A related open problem is the question about the possible existence of other such families in the context of Schmidt type partition numbers d k (n).
Many of the presented results were proven with the use of computer algebra. Nevertheless, in order to obtain more substantial mathematical insight, classical proofs, such as that one of Theorem 5, given in Section 6, would be desirable.