English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhartpolynomial are positive. Answering an Ehrhart positivity question posed onMathoverflow, Stanley provided an example of a non-Ehrhart-positive orderpolytope of dimension $21$. Stanley's example comes from a certain family oforder polytopes. In this paper, we study the Ehrhart positivity question onthis family of polytopes. By giving explicit formulas for the coefficients ofthe Ehrhart polynomials of these polytopes in terms of Bernolli numbers, wedetermine the sign of each Ehrhart coefficient of each polytope in the family.  As a consequence of our result, we conclude that for any positive integer $d\ge 21,$ there exists an order polytope of dimension $d$ that is not Ehrhartpositive, and for any positive integer $\ell$, there exists an order polytopewhose Ehrhart polynomial has precisely $\ell$ negative coefficients, whichanswers a question posed by Hibi. We finish this article by discussing theexistence of lower-dimensional order polytopes whose Ehrhart polynomials have anegative coefficient.

Stanley's non-Ehrhart-positive order polytopes