OEIS A000276: Two-Cycle Permutations With No Fixed Points

In this episode we explore A000276, the associated Stirling numbers of the first kind that count permutations of n with no fixed points and exactly two cycles. We unpack the defining count and the key formula a_n = n! × sum_{k=2}^{n-2} (1/k), showing how these integers refine permutation cycle structure. We’ll see how traditional tables look like stair-steps, and how a linear transformation reshapes them into a Pascal-like arithmetical triangle, revealing hidden order. We’ll also note that, unlike many Stirling numbers of the second kind, these do not form a Newton–Euler sequence, highlighting their distinctive divisibility and congruence behavior. Finally, we discuss applications in combinatorics and graph theory where counting cycle configurations with no fixed points matters, illustrating why these numbers matter beyond pure theory.

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