A Sneak-Peek Into Some Wonders Of Partitions
“Mathematics is the music of reason.” — James Joseph Sylvester, renowned mathematician
The Theory of Partitions is a beautiful area of mathematics and an active branch of number theory (the study of the integers and integer-valued functions) and combinatorics (concerned with counting, as a means as well as an end in obtaining results).
In simple terms, partitioning involves math problems with large numbers being split into smaller parts.
It can be understood better by means of an example. Say, there is a number 6. It can be partitioned in 11 unique ways viz., 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1. Here, we regard 5+1 as the same partition of 6 as 1+5.
Initiated by Euler, eminent minds such as J. J. Sylvester, S. Ramanujan, and G. H. Hardy made significant contributions to this field. Ramanujan’s advances opened doors for some of the most recent domains of mathematical (pure as well as applied) study and research. This article is a small attempt to throw light on some interesting deets about partitions!
Towards the Beginning
Herbert Wilf, a renowned mathematician, once said, “a generating function of a sequence is a clothesline on which you hang all the elements of the sequence.” Basically, if we want to carry all the clothes together, we just take the clothesline (hanger) and move from one place to another. It means that if we want to understand the properties of the sequence all at once, we can do so by studying the generating function of that sequence which carries all the information about its coefficients.
In the 18th century, Euler found a generating function for the partition function p(n), which simply counts the number of partitions of the positive integer n. There are quite many things that one can do with this function. A simple illustration can be made based on the example that we discussed earlier — the partitions of the number 6. If we consider the partitions that consist of only odd parts, they are four in number: 5+1, 3+3, 3+1+1+1, and 1+1+1+1+1+1. Next, if we consider the partitions in which the parts are not repeated, they are also four in number: 6, 5+1, 4+2, and 3+2+1. Now a question arises — is this just a coincidence or a property of every natural number? Well, it is not a coincidence according to Euler’s theorem which says — the number of partitions of n into odd parts always equal the number of partitions of n into distinct parts. It can be proved using generating functions.
All this eventually paved the way for what is known to the world as the theory of partitions. Partitions have a wide diversity of applications such as in nonparametric statistics (uses data that does not rely on numbers but on order or ranking of sorts), probability and statistics, particle physics, computer science, and group theory (studies the algebraic structures called groups) through Young Tableaux (a convenient way to describe the group representations of the symmetric and general linear groups and study their properties). Other areas include basic hypergeometric series, modular forms, and the latest, mock modular forms.
The Torchbearers
The Systemic Theory of Partitions begins with Leonhard Euler, an influential mathematician, physicist, astronomer, geographer, logician, and an engineer. Johann Carl Friedrich Gauss (mathematician and physicist) and Carl Gustav Jacob Jacobi (mathematician) also contributed to this area of study. The English mathematician, James Joseph Sylvester, played a crucial role in developing it further. Furthermore, Srinivasa Ramanujan and Godfrey Harold Hardy (renowned mathematicians) made some stunning advancements in this domain.
Talking about Contemporary Partition Theory, two of the major contributor figures are George Eyre Andrews (mathematician) and Freeman John Dyson (theoretical physicist and mathematician).
Hardy and Ramanujan thought of approaching partitions analytically. They asked Percy Alexander MacMahon to compute p(1), p(2),…p(200). His table was later extended by Hansraj Gupta upto p(300). Ramanujan analyzed MacMahon’s works, observed some patterns, and, based on them, gave what are now known as Ramanujan’s Partition Congruences. An expression for the generating function for the partition function associated with one of his formulas from this set, according to Hardy, was Ramanujan’s most beautiful identity!
The Tale of Rank and Crank
Introduced by Dyson, the rank of a partition is the number that is obtained by subtracting the number of parts from the largest part in that partition. This concept was presented in the context of a study on some of the partition congruences discovered by Ramanujan.
Dyson moved on to conjecture the crank of a partition, a statistic that would provide a combinatorial interpretation of the third congruence and yield a proper explanation of all three congruences. It was probably one of the first instances where a mathematical function was named before it was defined! Finally, this hypothetical concept was realized 44 years later by George Andrews and Frank Garvan. In 1987, there was a centenary to celebrate Ramanujan’s 100th birthday at the University of Illinois Urbana Champaign, at the end of which these two mathematicians collaborated together and by the end of the night, they were ready with the definition of the crank!
Trotting through Some Proofs and Theorems
Hardy-Ramanujan-Rademacher formula, Euler’s Pentagonal Number Theorem, and Franklin’s Proof of Euler’s PNT are some of the most critical developments in the field of partitions. It is very difficult, sometimes, to produce a bijection (a function between the elements of two sets, where each element of one set is paired with one and only one element of the other set) using combinatorial techniques. Often, these bijective proofs give insights into many new constructs and observations that would not have been possible otherwise.
There were some identities that were discovered and proved by mathematician Leonard James Rogers in 1894. These were later rediscovered by Ramanujan sometime around 1913 and came to be known as the Rogers-Ramanujan identities. These identities give product representations for what are called the Rogers-Ramanujan functions. It was because of Ramanujan that these earlier forgotten works became known to the world, which resulted in Rogers achieving unprecedented fame! Moreover, both the identities have interesting partition-theoretic interpretations that were proved combinatorially by Russian mathematician Issai Schur. Closely connected with the Rogers-Ramanujan functions is the Rogers-Ramanujan continued fraction. Taking a look at the special values of this continued fraction given by Ramanujan in his first letter to Hardy, the latter expressed, “…these formulas defeated me completely. I had never seen anything in the least like them before.”
These identities have connections with several areas of mathematics and science. These include Lie algebras, statistical mechanics (Rodney Baxter’s solution of the hard hexagon model), partition theory, modular forms, algebraic geometry, K-theory, conformal field theory, group theory, knot theory, transcendental number theory, orthogonal polynomials, probability, Kac-Moody, Virasoro, vertex, and double affine Hecke algebras, to name a few.
In the End…
Partitions form an integral as well as a wonderful part, not just of mathematics but its many allied disciplines as well.
In the words of J. J. Sylvester, “partitions constitute the sphere in which analysis lives, moves, and has its being; and no power of language can exaggerate or paint too forcibly the importance of this till recently almost neglected, but vast, subtle, and universally permeating element of algebraical thought and expression.”
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This article is based on one of the sessions (delivered by Atul Abhay Dixit, faculty in the Mathematics discipline) of the Virtual Seminar Series by IIT Gandhinagar. It is an online program started by the Institute in the wake of the current pandemic as a means to engage the people so that they can learn about a diversity of topics from the comfort of their homes, in an interesting manner. (The 4th article of this series can be found here. The 6th article is available here.)
