Integer Partitions

A partition of n is a way to write n as a sum of positive integers. p(5)=7, p(10)=42, p(100)=190,569,292. Hardy-Ramanujan formula: p(n) ~ exp(π√(2n/3)) / (4n√3). Young diagrams visualize partitions; conjugation = transpose. Ferrers shape and hook length formula for counting tableaux.

n10

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p(10) = 42

Partitions of 10

Generating function:
Σ p(n)xⁿ = Π 1/(1-xᵏ)

Hardy-Ramanujan (1918):
p(n) ~ eᵖⁱ√(2n/3) / 4n√3

Euler's pentagonal theorem:
Π(1-xⁿ) = Σ(-1)ᵏ x^ω(k)
ω(k) = k(3k-1)/2

Hook Length Formula

        For SYT of shape λ:

        f^λ = n! / Π h(x)

        h(x) = arm + leg + 1