All unimodal coefficient orders arise
Abstract
Exactly the unimodal permutations occur as coefficient orders of products of positive linear factors.
Theorem 1.1 (Answer to Question 4.6).
Lean statement: D5/S3/Combinatorics/RealRootedPermutations/Result.result
Proof. Machine-checked in Lean as D5/S3/Combinatorics/RealRootedPermutations/Result.result (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. T. Ball, D. Galvin, C. Hyry, and K. Weingartner (2022). Independent set and matching permutations. DOI: 10.1002/jgt.22724. URL: https://doi.org/10.1002/jgt.22724.
Commentary.
For every natural n and permutation of Fin(n+1), there are n+1 strictly positive real parameters whose nonconstant product coefficients weakly increase in that permutation order if and only if the permutation is unimodal. Value v denotes coefficient index v+1, so the constant coefficient is omitted. The definitions follow equations (1) and (4) of the cited article, and the conclusion answers Question 4.6 in Section 4 for polynomials of full degree.
LogConcave.unimodal_of_arises gives the necessary condition, including weak orders with a tie at the coefficient maximum. Realization.realize_strict gives the sufficient condition with strictly increasing coefficients, hence also a weak order. The realization uses integer exponent heights and a base larger than the number of subset terms. The classification of unweighted graph matching permutations is a separate question.