OEIS A000065: Partitions, Geometry, Graphs, and Beyond

We peel back the layers of A000065, the ‘minus one plus the number of partitions’ sequence, and watch it thread through geometry, graph theory, and even real-world systems. We’ll explore its geometric side: counting n-dimensional simplices with integer edge lengths and a fixed diameter, revealing how partitions shape the simplest building blocks of space. In graph theory, we’ll examine graphs with tree width n^2 and their near-tree-like structure, and how rooted trees of height two connect to algorithmic questions in data organization. The story extends to networks and supply chains: a 2013 Modrak–Martin framework uses A000065 to quantify topology complexity, offering a bridge from abstract partitions to real-world resilience and design. We’ll also touch on partitions viewed through alternative lenses—sums plus parts constraints—and finally glimpse a striking property of the complements of those near-tree graphs: a forbidden-substructure flavor that exposes an elegant, layered structure beneath the complexity. Join us as we uncover how this simple sequence quietly orchestrates a surprising chorus across mathematics and beyond.

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