On the real-rootedness of Ehrhart h*-polynomials of $(132,213)$-avoiding permutahedra
Abstract
Let Pd be the convex hull of the permutations of {1,…,d} that avoid the patterns $132$ and $213$, and let Hd(t) be its Ehrhart h*-polynomial. We prove that all zeros of Hd(t) are real and negative for 3≤ d≤1000 and for every d≥262; for d≤2 the polynomial is constant. Expanding Hd(t) in refined Eulerian polynomials does not suffice, since a coordinate is already negative at d=7. Let Ad,i(t) enumerate all permutations of {1,…,d} with first entry i by number of descents. For c≥0, we prove that every nonnegative linear combination of Ad,1,…,Ad,d and the adjacent differences Ad,r+1−cAd,r, 2≤ r≤ d−2, is real-rooted for all d≥4 simultaneously if and only if c≤2−√3. This reduces real-rootedness to the positivity of explicit weighted tails of the coordinates. An exact certificate checks these inequalities for d≤1000, and coefficient estimates based on Darroch's theorem on the mode of a sum of Bernoulli variables prove them for d≥262; the range 1001≤ d<262 remains open. Along the way we identify Pd, up to lattice equivalence, with twice a chain-poset permutahedron and with the Newton polytope of a discriminant, which transfers the known cube structure and volume of these models to Pd. We also show that the coefficient of td−1 in Hd(t) counts strong tournament score sequences for d≥3, confirming a conjecture of Rehberg in this range.
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BibTeX
@misc{castro2026realrootedness,
title = {{On the real-rootedness of Ehrhart $h^*$-polynomials of $(132,213)$-avoiding permutahedra}},
author = {Pedro M. M. de Castro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06096},
eprint = {2609.06096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06096},
}
RIS
TY - UNPB
TI - On the real-rootedness of Ehrhart h^*-polynomials of $(132,213)$-avoiding permutahedra
AU - Pedro M. M. de Castro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.06096
UR - https://arxiv.org/abs/2609.06096
AB - Let P_d be the convex hull of the permutations of {1,…,d} that avoid the patterns $132$ and $213$, and let H_d(t) be its Ehrhart h^*-polynomial. We prove that all zeros of H_d(t) are real and negative for 3≤ d≤1000 and for every d≥2^62; for d≤2 the polynomial is constant. Expanding H_d(t) in refined Eulerian polynomials does not suffice, since a coordinate is already negative at d=7. Let A_d,i(t) enumerate all permutations of {1,…,d} with first entry i by number of descents. For c≥0, we prove that every nonnegative linear combination of A_d,1,…,A_d,d and the adjacent differences A_d,r+1−cA_d,r, 2≤ r≤ d−2, is real-rooted for all d≥4 simultaneously if and only if c≤2−√3. This reduces real-rootedness to the positivity of explicit weighted tails of the coordinates. An exact certificate checks these inequalities for d≤1000, and coefficient estimates based on Darroch's theorem on the mode of a sum of Bernoulli variables prove them for d≥2^62; the range 1001≤ d<2^62 remains open. Along the way we identify P_d, up to lattice equivalence, with twice a chain-poset permutahedron and with the Newton polytope of a discriminant, which transfers the known cube structure and volume of these models to P_d. We also show that the coefficient of t^d−1 in H_d(t) counts strong tournament score sequences for d≥3, confirming a conjecture of Rehberg in this range.
ER -