OEIS A000070: Partitions, Ferrers diagrams, one-transitions, and cycle subgraphs

We dive into A000070, the cumulative partition-counting sequence. Beyond counting partitions of 0 through n, the entry reveals rich structure: Ferrers diagrams that illuminate the step to n+1, the poset of one-transitions between partitions, and a surprising graph-theory angle as the count of unlabeled subgraphs of the n-cycle. A compact tour of how a simple sequence connects number theory, combinatorics, and graph theory.

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