OEIS A000041: Partition numbers

Join us as we explore A000041, the partition numbers p(n): the number of ways to write n as a sum of positive integers, disregarding order. We trace their appearances across math—from conjugacy classes and irreducible representations of the symmetric group S_n to the classification of abelian groups of order n, and even to counting certain rooted trees of height at most 2. We also see connections to sigma-algebras, and we discuss the generating function ∏_{k≥1} (1−x^k)^{−1}, the Hardy–Ramanujan asymptotic, log-concavity, and Benford's law. We close with computational challenges and open questions, including Sun's conjecture that p(n) is never a perfect power for n>2, n≠6.

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