MacMahon's Partition Research Articles

MacMahon's partition analysis XV: Parity

We apply the methods of partition analysis to partitions in which the parity of parts plays a role. We begin with an in-depth treatment of the generating function for the partitions from the first Göllnitz-Gordon identity. We then deduce a Schmidt-type theorem related to the false theta functions. We also consider: (1) position parity, (2) partitions with distinct even parts, (3) partitions with distinct odd parts. One of the corollaries of these last considerations is a new interpretation of Hei-Chi Chan's cubic partitions. A second part of our article is devoted to the algorithmic derivation of identities and arithmetic congruences related to the generating functions considered in part one, including cubic partitions. To this end, Smoot's implementation of Radu's Ramanujan-Kolberg algorithm is used. Finally, we give a short description which explains how to use the Omega package to derive special instances of the results of part one.

MacMahon's partition analysis XIV: Partitions with n copies of n

We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.

Elementary proofs of infinitely many congruences for k-elongated partition diamonds

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers dk(n) produced by summing the links of k-elongated plane partition diamonds of length n is given by (q2;q2)∞k(q;q)∞3k+1 for each k≥1. A significant portion of their recent paper involves proving several congruence properties satisfied by d1,d2 and d3, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions dk for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.

Diagonal Hooks and a Schmidt-Type Partition Identity

In a recent paper of Andrews and Paule, several Schmidt-type partition identities are considered within the framework of MacMahon's Partition Analysis. Following their work, we derive a new Schmidt-type identity concerning diagonal hooks of partitions. We provide an analytic proof based on MacMahon's Partition Analysis and a combinatorial proof through an involution on the set of partitions. We also establish connections between Schmidt-type distinct partitions and partitions with nonpositive and negative cranks.

Schmidt type partitions

Recently, Andrews and Paule studied Schmidt type partitions using MacMahon's Partition Analysis and obtained various interesting results. In this paper, we focus on the combinatorics of Schmidt type partition theorems and characterize them in a general and refined form. In addition, we also present some overpartition analogues of Schmidt type partition theorems.

Extending Putzer's representation to all analytic matrix functions via omega matrix calculus

We show that Putzer's method to calculate the matrix exponential in [28] can be generalized to compute an arbitrary matrix function defined by a convergent power series. The main technical tool for adapting Putzer's formulation to the general setting is the omega matrix calculus; that is, an extension of MacMahon's partition analysis to the realm of matrix calculus and the method in [6]. Several results in the literature are shown to be special cases of our general formalism, including the computation of the fractional matrix exponentials introduced by Rodrigo [30].Our formulation is a much more general, direct, and conceptually simple method forcomputing analytic matrix functions. In our approach the recursive system of equations the base for Putzer's method is explicitly solved, and all we need todetermine is the analytic matrix functions.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/97/abstr.html

MacMahon Partition Analysis: A discrete approach to broken stick problems

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for n≥k≥3, the probability that a k-gon cannot be formed from a stick broken into n parts is given by n! over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.

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.

A Euclid style algorithm for MacMahon's partition analysis

Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but “bottlenecks” always arise when dealing with complex problems. While in computational geometry, Barvinok's important result asserts the existence of a polynomial time algorithm when the dimension is fixed. However, the implementation by the LattE package of De Loera et al. does not perform well in many situations. By combining excellent ideas in the two fields, we generalize Barvinok's result by giving a polynomial time algorithm for MacMahon's partition analysis in a suitable condition. We also present an elementary Euclid style algorithm, which might not be polynomial but is easy to implement and performs well. As applications, we contribute the generating series for magic squares of order 6.

CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS

Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.

MacMahon partition analysis and the Poincaré series of the algebras of invariants of ternary and quaternary forms

By using MacMahon partition analysis technique, the Poincaré series for the algebras of invariants of the ternary and quaternary forms of small degrees are calculated.

COMBINATORIAL PROOF OF ONE CONGRUENCE FOR THE BROKEN 1-DIAMOND PARTITION AND A GENERALIZATION

In one of their recent collaborative papers, Andrews and Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of diamond shape. They prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. In this note, we reprove this congruence by constructing an explicit way to group partitions. Then we keep the essence of the method and manage to apply it to a different kind of plane partitions to get more general results and several other congruences.

MacMahon's partition identity and the coin exchange problem

One of MacMahon's partition theorems says that the number of partitions of n into parts divisible by 2 or 3 equals the number of partitions of n into parts with multiplicity larger than 1. Recently, Holroyd has obtained a generalization. In this short note, we provide a bijective proof of his theorem.

Further developments in MacMahon's partition analysis (abstract only)

P.A. MacMahon developed a calculus for producing generating functions of partitions constrained by linear inequalities relating the various parts of the partitions. Recently we have used MacMahon's method to find new classes of partitions whose generating functions are modular forms (Acta Arith., 126(2007), 281-294), and we have retrieved MacMahon's original project which was to prove his generating function formula for plane partitions. We shall report briefly on these topics and indicate new areas for exploration. In particular, we shall consider the modification of MacMahon's methods in which the inequalities relating the various parts of the partition take into account congruence constraints on the parts.

MacMahon's partition analysis XII: Plane partitions

MacMahon developed partition analysis as a calculational and analytic method to produce the generating function for plane partitions. His efforts did not turn out as he had hoped, and he had to spend nearly twenty years finding an alternative treatment. This paper provides a detailed account of our retrieval of MacMahon’s original project. One of the key results obtained with partition analysis is an extension of a theorem of Gansner which generalizes Stanley’s famous trace theorem. This is the twelfth paper in this series on MacMahon’s partition analysis. It has been our belief from the beginning that MacMahon’s ideas could be best exploited by computer implementation, and that was the genesis of our partition analysis project. Our algorithmic version of MacMahon’s method has been implemented in the form of the Mathematica package Omega which is freely available via the web; see [20]. In the back of our minds was always MacMahon’s melodramatic experience with his own invention. He created partition analysis solely to treat the generating functions associated with various classes of plane partitions. Plane partitions are two-dimensional arrays of non-negative integers which are weakly decreasing in rows and columns; for a formal definition, including the notion of k-trace (1.3), see below. This specific project failed, and in this paper we shall retrieve MacMahon’s original project and obtain, using only partition analysis, an extension, namely Theorem 5.4, of a general plane partition theorem originally due to Gansner [13, Theorem 4.2]. For further remarks on plane partition history, in particular, on how partition analysis has led us to a rediscovery and to an alternative proof of Gansner’s theorem, we refer the interested reader to [4]. The initial stage of MacMahon’s investigations is chronicled by him in [16], where he refines the study of partitions and compositions of multipartite numbers into the theory of plane partitions. MacMahon [16, p. 658] first stated as an unproven assertion that the generating function for plane partitions is, in fact,

Constructing all magic squares of order three

We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.

On recent congruence results of Andrews and Paule for broken k-diamonds

In one of their most recent works, George Andrews and Peter Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of hexagons. In the process, they prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. Our first goal in this note is to reprove this congruence by explicitly finding the generating function in question. We then prove one of the conjectures posed by Andrews and Paule as well as a number of congruences not mentioned by them. All of our results follow from straightforward generating function manipulations.

MacMahon's partition analysis XI: Broken diamonds and modular forms

MacMahon's partition analysis XI: Broken diamonds and modular forms

A Fast Algorithm for MacMahon's Partition Analysis

This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

MacMahon's Partition Analysis VI: A New Reduction Algorithm

In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. By developing the Omega package we have shown that Partition Analysis is ideally suited for being supplemented by computer algebra methods. The object of this paper is to present a significant algorithmic improvement of this package. It overcomes a problem related to the computational treatment of roots of unity. Moreover, this new reduction strategy turns out to be superior to "The Method of Elliott" which is described in MacMahon's book. In order to make this article as self-contained as possible we give a brief introduction to Partition Analysis together with a variety of illustrative examples. For instance, the generating function of magic pentagrams is computed.