Séminaire Lotharingien de Combinatoire, 95B.55 (2026), 12 pp.
Yakob Kahane
On Two Open Problems on Ehrhart Polynomials of Fences
Abstract.
We show how the coefficients of the order polynomial of a poset can be computed as a weighted sum over all decompositions into convex subposets. Using this result, we provide an explicit combinatorial interpretation of the coefficients of the order polynomial of a fence poset and conjecture a generalization to skew shape posets. Moreover, we use the decomposition into convex subposets to provide a combinatorial interpretation of the coefficients of the Ehrhart polynomial evaluated at t-3/2 associated with the zigzag posets. This allows us to express its generating series as a Hadamard product of two explicit series. Using this last result, we give an alternative proof of the second part of the Watanabe–Yoshida conjecture from 2005
by establishing an explicit lower bound on the evaluation of these shifted Ehrhart polynomials.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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