Generating Functions for Integer Partitions Explained

Euler's Pentagonal Number Theorem provides a recurrence relation for the number of integer partitions of a positive integer n, which states that p(n) = ∑_{k} (-1)^{k+1} p(n - g_k), where g_k = k(3k-1)/2 are the generalized pentagonal numbers and the sum is taken over all k such that g_k ≤ n; this theorem allows the computation of partition numbers using only smaller partition numbers, and it also reveals that the difference between the number of partitions of n into an even number of distinct parts and the number into an odd number of distinct parts is equal to (-1)^m if n is a generalized pentagonal number, and zero otherwise.

Euler's Pentagonal Number Theorem provides a recursive formula for calculating the number of integer partitions of a positive integer n, where p(n) = ∑(-1)^m p(n - u_m) for all generalized pentagonal numbers u_m = m(3m±1)/2, meaning the number of partitions of n can be computed using previous partition counts that differ by generalized pentagonal numbers; this theorem also establishes that the difference between the number of partitions of n into an even number of distinct parts and the number into an odd number of distinct parts equals (-1)^m if n is a generalized pentagonal number, and zero otherwise.

Euler's recurrence formula for the partition function p(n) is given by p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + p(n-12) + p(n-15) - ..., where the terms involve generalized pentagonal numbers (1, 2, 5, 7, 12, 15, 22, 26, 35, 40, ...). This formula allows calculation of the number of ways to partition an integer n into sum of positive integers by recursively computing values from smaller integers.

Euler's pentagonal number theorem provides a recursive formula for calculating partition numbers, which count the number of ways to write a positive integer as a sum of positive integers. The theorem states that the difference between the number of even and odd distinct partitions of an integer is zero for non-pentagonal numbers, but equals ±1 for pentagonal numbers. This leads to the recursive formula p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + p(n-12) + p(n-15) - ..., where the positions of the plus and minus signs follow the pentagonal number sequence generated by the formula n(3n±1)/2. This elegant result, discovered by Euler and later visualized by Fabian Franklin in 1881, connects seemingly unrelated areas of mathematics including number theory, combinatorics, and prime detection.

Euler's Pentagonal Number Theorem states that the generating function for integer partitions, P(x) = ∏_{k=1}^∞ (1 - x^k)^(-1), can be expressed as a sum involving pentagonal numbers: P(x) = ∑_{n=-∞}^∞ (-1)^n x^(n(3n-1)/2). This theorem provides a powerful recurrence relation for computing partition numbers: p(n) = ∑ (-1)^k p(n - g_k), where g_k = k(3k-1)/2 are the pentagonal numbers (1, 2, 5, 7, 12, 15, 22, 26, 35, 40, ...). The theorem was conjectured by Euler in 1741 but not proven until much later. The proof involves a combinatorial bijection between partitions into distinct parts and partitions into odd parts, using Maya diagrams to establish the connection between the product and sum forms of the generating function.

Three fundamental partition identities form the foundation of this lecture. Euler's first identity (A) generates all integer partitions using the series and product formula. Euler's second identity (B) counts partitions into distinct parts where each part differs by at least one from the next. The Rogers-Ramanujan identity (C), published by L.J. Rogers in 1894 and independently conjectured by Ramanujan in 1914, states that the number of partitions where parts differ by at least two equals the number of partitions into parts congruent to 1 and 4 modulo 5. These identities demonstrate how different types of partitions can be counted in two equivalent ways through series-product relationships.

Partition identities are mathematical equalities between two types of partition counts: the 'product side' (partitions obeying congruence conditions) and the 'sum side' (partitions obeying difference conditions and initial conditions). Researchers use computer algebra systems to generate potential sum sides and apply Euler's algorithm to identify which ones produce elegant product sides, rediscovering known identities like Euler's odd-distinct parts equivalence and Rogers-Ramanujan identities while also discovering new ones such as Capparelli's identity.

This section develops advanced partition theory: partitions with minimum difference 2 between parts require sophisticated variable transformations yielding generating functions involving q^{k²} and q^{k²+k}. The Rogers-Ramanujan identities connect these difference-constrained partitions to modular partition classes: partitions with minimum difference 2 correspond to parts ≡1,4 mod 5, while those additionally requiring parts >1 correspond to parts ≡2,3 mod 5. These identities represent profound connections between seemingly unrelated partition conditions.

Euler's identity states that for any non-negative integer r, the number of partitions of r into distinct parts equals the number of partitions of r into odd parts. For example, for r=5, there are 3 partitions into distinct parts (5, 4+1, 3+2) and also 3 partitions into odd parts (5, 3+1+1, 1+1+1+1+1). This identity is proven by showing that their generating functions are equal: ∏_{i=1}^∞ (1 + x^i) = ∏_{i=1}^∞ 1/(1 - x^{2i-1}).

Ramanujan discovered what are now called the Rogers-Ramanujan identities, which reveal fascinating properties of partitions. These mysterious formulas show that certain restricted classes of partitions have matching totals. For example, when restricting partitions of 11 in two different ways—those where all parts differ by at least two, and those where each part ends with 1, 4, 6, or 9—there are exactly seven partitions in each restricted class. The Rogers-Ramanujan identities imply that these totals match for the partitions of any number n. These identities continue to puzzle and inspire mathematicians today, demonstrating how Ramanujan's work produced formulas that still leave mathematicians speechless.

The Hardy-Ramanujan theorem gives an asymptotic formula for p(n): log p(n) grows like √n, so p(n) ~ e^(√n). More precisely, p(n) ~ (1/(4n√3)) * e^(π√(2n/3)). The circle method also provides a divergent series that, when truncated appropriately, gives p(n) with error at most 0.5.

The Hardy-Ramanujan asymptotic formula for ordinary partitions is log p(n) ~ π√(2n/3), with p(n) ~ (4n√3)/(3π) × exp(π√(2n/3)). This formula gives the exponential growth rate. However, for partitions into squares, the corresponding formula is much weaker, giving only the leading term with error of order 1/√n. The method cannot be improved by looking at other roots of unity because the function lacks modularity. This limitation motivated the speaker's research into finding better asymptotic formulas.

The Hardy-Ramanujan asymptotic formula p(n) ≈ (1/(4n√3)) × exp(π√(4n/3)) provides increasingly accurate approximations for large n, with about 4.6% error at n=100. Ramanujan's full series expansion enables calculation of partition numbers for n=200 to the nearest integer using only five terms. Partition numbers reach astronomical sizes—numbers around 10^20 have partition counts with approximately 11 billion digits. Beyond pure mathematics, partition theory applies to card shuffling combinatorics and statistical mechanics, where partition functions describe energy distribution among particles.

G.H. Hardy and Srinivasa Ramanujan developed an asymptotic formula for the partition function p(n). The leading term is approximately exp(π√(2n/3)). More precisely, p(n) ≈ (1/(4n√3)) × exp(π√(4n/3)). This formula is derived using the circle method, which analyzes the behavior of the generating function near roots of unity. Remarkably, truncating this infinite series after about the fourth root of n terms gives an exact integer value for p(n).

The Hardy-Ramanujan collaboration produced the famous asymptotic formula for partition functions, representing one of mathematics' most celebrated partnerships. A partition is a way of writing an integer as a sum of positive integers—for n=200, there are approximately three trillion partitions. The formula uses a series involving complex analysis, with the critical insight that the term n - 1/24 must be included. Hardy provided technical mastery of complex variables to establish error bounds, while Ramanujan provided the brilliant conjectures. Littlewood described it as 'a singularly happy collaboration of two men of quite unlike gifts in which each contributed the best, most characteristic, and most fortunate work that was in him.' This demonstrates how complementary strengths—Ramanujan's intuition combined with Hardy's rigor—produce landmark mathematical results.

For bosons (identical particles that can occupy the same quantum state), only the unique states are counted without considering permutations. This reduces the number of terms from 9 to 6. The partition function becomes Z = 1 + e^(-2e/kT) + e^(-6e/kT) + e^(-4e/kT) + e^(-3e/kT). Using β = 1/kT, this simplifies to Z = 1 + e^(-2eβ) + e^(-6eβ) + e^(-4eβ) + e^(-3eβ).

The grand canonical ensemble formalism is applied to an ideal Bose gas by considering a single quantum state as the system and all other states as the environment. For a single quantum state with energy ε, the grand partition function is Z = Σ_n e^(-n(ε-μ)/k_B T) = 1/(1 - e^(-(ε-μ)/k_B T)), where n = 0, 1, 2, ... is the occupation number. This geometric series converges only if e^(-(ε-μ)/k_B T) < 1, which requires μ < ε for all states. The average occupation number is n(ε) = 1/(e^((ε-μ)/k_B T) - 1), the Bose-Einstein distribution function. The total number of particles is N = Σ_i 1/(e^((ε_i-μ)/k_B T) - 1). The function g_{3/2}(z) has a singularity at z = 1, which corresponds to the onset of Bose-Einstein condensation, where a macroscopic number of particles occupy the ground state.

Bose-Einstein statistics applies to indistinguishable bosons (like photons) that can occupy the same state in unlimited numbers. The thermodynamic probability is W_BE = ∏((n_i + g_i - 1)! / (n_i! (g_i - 1)!)). The partition function Z = Σe^(-βE_i) encodes all system information, with thermodynamic properties derived from it. Understanding these concepts allows solving any statistical mechanics problem.

The partition function serves as a generating function for calculating ensemble averages. Derivatives of log Z yield statistical quantities: ∂log Z/∂μ = β⟨N⟩, ∂log Z/∂β = -⟨E - μN⟩, and ∂log Z/∂V = βP. Higher-order derivatives generate fluctuations: ∂²log Z/∂β² = ⟨E²⟩ - ⟨E⟩² = ΔE². This powerful formalism allows calculation of energy, pressure, particle number, and their fluctuations without explicit summation over all microstates.

This section traces the progression from basic combinatorial objects to complex statistical mechanics models. Integer compositions are ordered sums with simple formulas, while integer partitions are unordered with no closed-form count. Young diagrams visualize partitions as left-justified boxes. Plane partitions extend this to 2D grids with weakly decreasing entries, representing dimer models on hexagonal grids. Euler's generating function for partitions is ∏_{n≥1} 1/(1-q^n). The Hillman-Grassl bijection transforms reverse plane partitions into arrays with independent entries, enabling generating function computations as products over hook weights. This algebraic combinatorics framework connects to statistical mechanics through plane partitions modeling dimer configurations.