Eventual Strict Log-Concavity of Weighted Linear Hypercombs

Abstract

We prove that the coefficient sequence of a weighted linear hypercomb is strictly log-concave at every interior index once its length is sufficiently large, for each fixed pair of positive real weights. Integer weights recover strong independent-set sequences with possibly different spine-edge and tooth sizes. In the uniform case, the result eliminates the possible interval of nonmonotonicity in the theorem of Galvin and Sharpe for every fixed uniformity and all sufficiently large lengths. The proof combines fixed-index coefficient estimates, tilted Fourier integrals and polynomial reversal; a weighted-variance identity establishes the required nondegeneracy. We also show that no length threshold works for all positive weights. When the two weights are equal and tend to zero while their product with the length tends to a positive constant, we identify a boundary scaling limit. The limiting coefficient sequence is log-concave exactly when the scaling parameter is at least an explicit algebraic constant, approximately 0.1879701949757, determined by the fifth coefficient inequality. This constant also bounds from below the lower limit of the product of the common weight and the least eventual length threshold as the weight tends to zero. Lean proofs accompany the main coefficient theorem, the threshold obstruction and the exact transition for the recursively defined limit sequence. The general scaling limit and generating-function identification are established in the written proof.

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