Integer partition

An integer partition writes a nonnegative integer as a sum of positive integers, with order ignored. The empty sum is the unique partition of zero.

Integer composition: An ordered sequence of positive integers whose sum is a specified integer. Compositions count the same summands as partitions but treat different orders as distinct.
Partition generating function: A power series whose coefficient of each power counts partitions satisfying specified rules. Its product formula encodes every possible multiplicity of each positive part.
Leonhard Euler: An eighteenth-century mathematician whose work established foundational results across number theory and analysis. Euler derived the generating-function identity that became central to partition counting.
Representation theory of the symmetric group: The study of linear representations of permutations of a finite set. Partitions index the irreducible representations of symmetric groups.
Hardy–Ramanujan asymptotic formula: An asymptotic formula estimating the partition number p(n) as n grows. It captures the rapid growth of the number of partitions using an exponential expression.
Partition number: The number of integer partitions of a specified nonnegative integer. It turns the set of partitions of each integer into a sequence of counts.
Euler's pentagonal number theorem: An identity expressing the reciprocal partition product as a series with pentagonal-number exponents. It yields a sparse recurrence for computing unrestricted partition numbers.
Pentagonal number theorem: An identity expanding an infinite product as an alternating sum indexed by generalized pentagonal numbers. Euler's identity links the product for partitions to a sparse pentagonal-number series.
Young tableau: A filling of a Young diagram's cells with entries subject to ordering conditions. Tableaux built from partition shapes encode representations and combinatorial identities.
Rademacher's formula: An exact convergent series for the partition number p(n). It refines Hardy and Ramanujan's approximation into an exact formula.

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