arXiv, preprint, v1
Research account
Ehrhart h*-polynomials of (132,213)-avoiding permutation polytopes: A repair cone and eventual real-rootedness
Lattice-point counts connect geometry with combinatorics: dilating an integral polytope produces a polynomial counting function, whose generating series has an h*-polynomial as its numerator. When all zeros of this numerator are real, its coefficients satisfy strong regularity properties, including log-concavity and unimodality. We study this question for a family introduced by Davis and Sagan, the convex hulls of permutations of length d avoiding the patterns 132 and 213.
A change to deficit coordinates makes the geometry explicit. The vertices are indexed by cuts of a path, and the face lattice is that of a cube, as illustrated for d = 3 and d = 4. This description also gives explicit lattice-point and volume formulas.
The main difficulty is that a natural expansion in refined Eulerian polynomials has negative coefficients already at d = 7. We construct a cone of adjacent signed differences that still preserves real-rootedness, with the sharp uniform repair parameter 2 − √3. The resulting criterion proves real-rootedness for 3 ≤ d ≤ 1000 by exact computation and for every d ≥ 2⁷² by an analytic argument. The intermediate range remains open.
The preprint is available on arXiv. Exact data through d = 1000 and instructions for reconstructing them from the paper's pseudocode are available in the companion repository.