5.1 Integer Partitions and Their Properties

Integer partitions are a key concept in algebraic combinatorics. They involve breaking down positive integers into sums of other positive integers, ignoring order. This seemingly simple idea has deep connections to other areas of math and physics.

Studying integer partitions reveals fascinating properties and relationships. From generating functions to asymptotic behavior, these patterns offer insights into number theory, representation theory, and even statistical mechanics. Understanding partitions is crucial for grasping the broader landscape of combinatorics.

Integer partitions and combinatorics

Definition and notation

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• An integer partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers, where the order of the summands does not matter
• The number of partitions of $n$ is denoted by $[p(n)](https://www.fiveableKeyTerm:p(n))$
  • For example, $p(4) = 5$ because 4 can be partitioned in five distinct ways: $4$, $3+1$, $2+2$, $2+1+1$, and $1+1+1+1$

Role in algebraic combinatorics

• Integer partitions are a fundamental concept in algebraic combinatorics
  • They are connected to various other combinatorial objects (Young tableaux, symmetric functions)
  • Have applications in number theory, representation theory, and statistical mechanics
• The study of integer partitions involves understanding their properties, enumeration, and relationships with other mathematical structures
• Generating functions are a powerful tool in the study of integer partitions
  • They can be used to derive identities and study asymptotic behavior

Properties of integer partitions

Partition theorems

• Euler's partition theorem: The number of partitions of $n$ into distinct parts is equal to the number of partitions of $n$ into odd parts
• Conjugate partition theorem: The number of partitions of $n$ into exactly $k$ parts is equal to the number of partitions of $n$ with largest part equal to $k$

Generating functions and identities

• The generating function for the number of partitions of $n$ is given by the infinite product $\prod_{k=1}^{\infty} (1 - x^k)^{-1} = (1 + x + x^2 + ...)(1 + x^2 + x^4 + ...)(1 + x^3 + x^6 + ...)$
• Ramanujan's congruences state that for any non-negative integer $k$:
  • $p(5k + 4) \equiv 0 \pmod{5}$
  • $p(7k + 5) \equiv 0 \pmod{7}$
  • $p(11k + 6) \equiv 0 \pmod{11}$

Asymptotic behavior

• The Hardy-Ramanujan-Rademacher formula provides an exact formula for $p(n)$ as an infinite sum involving Bessel functions and Kloosterman sums
• The asymptotics of $p(n)$ are given by the Hardy-Ramanujan asymptotic formula:
  • $p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left(\pi \sqrt{\frac{2n}{3}}\right)$ as $n \to \infty$

Generating integer partitions

Recursive methods

• Integer partitions can be generated recursively using the recurrence relation:
  • $p(n, k) = p(n-k, k) + p(n-1, k-1)$, where $p(n, k)$ denotes the number of partitions of $n$ into parts of size at most $k$

Graphical representations

• Ferrers diagram: A graphical representation of a partition consisting of rows of dots, where the number of dots in each row corresponds to the size of each part in the partition
  • Ferrers diagrams can be used to visualize and study properties of partitions
• Young tableau: Obtained by filling the Ferrers diagram with positive integers that are strictly increasing in each row and weakly increasing in each column
  • The number of Young tableaux of a given shape is related to the dimension of certain representations of the symmetric group

Combinatorial identities

• Euler pentagonal number theorem: $\prod_{k=1}^{\infty} (1 - x^k) = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + ...$, where the exponents are the generalized pentagonal numbers
• Robinson-Schensted-Knuth (RSK) correspondence: A bijection between matrices with non-negative integer entries and pairs of Young tableaux of the same shape
  • Has applications in the study of longest increasing subsequences and the representation theory of the symmetric group

Integer partitions vs other objects

Representation theory

• The number of irreducible representations of the symmetric group $S_n$ is equal to the number of integer partitions of $n$
• Frobenius characteristic map: A ring isomorphism between the ring of symmetric functions and the character ring of the symmetric group
  • Relates Schur functions (symmetric functions) to the irreducible characters of $S_n$

Partition algebras and statistical mechanics

• Partition algebras, introduced by Martin and Jones, are finite-dimensional associative algebras that arise in the study of the Potts model in statistical mechanics and the representation theory of the symmetric group
  • The dimension of the partition algebra is given by the Bell number, which counts the number of partitions of a set

Combinatorial identities and mathematical physics

• Kronecker coefficients: Appear in the tensor product decomposition of irreducible representations of the symmetric group
  • Related to the Kronecker product of Schur functions and the Littlewood-Richardson coefficients
• Rogers-Ramanujan identities: Identities involving infinite products and infinite sums, with a combinatorial interpretation in terms of integer partitions with certain restrictions on the sizes of parts and the differences between consecutive parts
• Nekrasov-Okounkov formula: Relates the hook lengths of partitions to the Seiberg-Witten prepotential in supersymmetric gauge theories
  • Provides a connection between integer partitions and mathematical physics