The partition numbers \(p(n)\) count the ways to write \(n\) as a sum of positive integers, ignoring order: \(4 = 4 = 3{+}1 = 2{+}2 = 2{+}1{+}1 = 1{+}1{+}1{+}1\), so \(p(4) = 5\). The sequence of values is 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135.
Partitions are a cornerstone of combinatorics and number theory. Euler found the generating function and the pentagonal-number recurrence used to compute them here; Hardy and Ramanujan derived an astonishing asymptotic formula, later made exact by Rademacher; and Ramanujan discovered the beautiful congruences \(p(5k+4) \equiv 0 \pmod 5\), \(p(7k+5) \equiv 0 \pmod 7\), and \(p(11k+6) \equiv 0 \pmod{11}\).