OEIS A000137: Series Parallel Numbers

Today we dive into A000137, the sequence counting series-parallel graphs by edge count. We’ll uncover its link to A000084 (series-parallel networks) through a generating-function relationship: the generating function for A000137 is x(1 + s)/(1 − s), where s is the generating function for A000084. We’ll discuss how the recursive, decomposable structure of series-parallel graphs enables dynamic-programming approaches that tame many otherwise NP-complete problems (such as maximum independent set in these graphs). We’ll also explore what the growth of A000137 reveals about complexity, and why these graphs matter in circuit design and algorithm analysis, all while highlighting how the OEIS stitches together surprising connections across mathematics.

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