OEIS A00258: Two-Level Partitions and Three-Level Labeled Rooted Trees

In this episode we dive into OEIS A00258. We unpack its two-stage counting: partitions of a set into blocks, then partitions of those blocks, equivalently three-level labeled rooted trees with N leaves. We’ll uncover the key identity A_N = sum_{k=0}^N S(N,k) Bell(k), linking Stirling numbers of the second kind and Bell numbers, and connect the iterated exponential generating function XPX11 at the heart of the definition. We’ll explore the history (M2932, M1178) and the OEIS community’s ongoing updates led by Neil Sloan, and we’ll trace surprising connections to class partition algebras, Hopf algebras, and even quantum physics. Practical takeaways include intuition for the combinatorial meaning and tips for computing these numbers in common math software.

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