Counts of maximal matchings in a graph need not be log-concave

The numbers of inclusion-maximal matchings of each size in a graph need not form a log-concave sequence. We give an explicit counterexample on 76 vertices: the join of six disjoint edges with two disjoint cliques of order 32. Its maximal matchings have sizes 35 through 38, and the square of the coefficient at size 36 is smaller than the product of its two neighbors. This answers the general log-concavity question posed by Došlić and Short and by Došlić and Zubac. The proof uses the independent uncovered-set viewpoint discussed by Došlić and Short, retaining the number of perfect matchings for each uncovered set. For the stated graph, this reduces the coefficient calculation to a short recurrence on residual graphs. A second computation evaluates an explicit factorial sum, and both use exact integer arithmetic. The result concerns log-concavity for arbitrary graphs; the displayed coefficient sequence is unimodal. This work has not yet been reviewed by experts. The proofs, the code and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.