OEIS A00261: Beads, Necklaces, and Permanents

We dive into OEIS A00261, a rapidly growing sequence defined by a two-term recurrence with initial terms a1=0 and a2=1. It shows up in surprising combinatorial ways: (i) as the permanent of a specially structured 0‑1 matrix, counting certain perfect matchings; (ii) as a beads‑and‑cords counting model where n labeled beads are split between two kinds of objects—necklaces (excluding single‑bead necklaces) and three indistinguishable cords—combined via an exponential convolution that ties together derangements and cord arrangements. We’ll unpack the recurrence, walk through a concrete n=4 example (where a6=465), and explore how these distinct viewpoints connect algebra, graph theory, and even physics‑inspired diagram counting.

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