OEIS A000198: Automorphisms of tournaments

We explore A000198, the number of automorphisms of tournaments with n labeled vertices—a fascinating intersection of group theory, graph theory, and combinatorics. A tournament is a complete directed graph (an orientation of every edge). An automorphism is a relabeling of the vertices that preserves all edge directions, so Aut(T) is a subgroup of the symmetric group Sn. There is no simple closed form for A000198; terms are computed recursively from prior terms, with roots in work by Alspach and Berggren among others. We discuss why the sequence grows and what it reveals about symmetry in combinatorial structures.

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