Skew shapes, Ehrhart positivity and beyond

TL;DR

This work proves that the order polynomials $\Omega(P_{\lambda/\mu};t)$ of cell posets of arbitrary skew shapes have nonnegative coefficients, linking this positivity to Ehrhart positivity via order polytopes. The authors develop a general coproduct framework for $\Omega(P;t)$ and a meta-positivity principle that reduces the problem to the linear term, then apply Kreweras determinant analyses to skew shapes to establish the main result. Consequences include Ehrhart positivity for shard polytopes and for a broad family of matroid base polytopes (snake matroids and their direct sums with loops/coloops), as well as positivity for fence/zig-zag/circular-fence posets and several cylindric/shifted generalizations. The paper also connects to Schubert calculus, Macdonald identities, and a spectrum of conjectures, offering a unifying positivity mechanism across skew-shape combinatorics and related geometric objects. Overall, the results illuminate deep positivity phenomena in order polytopes, plane partitions, and associated polytopes, while outlining clear directions for geometric proofs and broader extensions.

Abstract

A classical result by Kreweras (1965) allows one to compute the number of plane partitions of a given skew shape and bounded parts as certain determinants. We prove that these determinants expand as polynomials with nonnegative coefficients. This result can be reformulated in terms of order polynomials of cell posets of skew shapes, and explains important positivity phenomena about the Ehrhart polynomials of shard polytopes, matroids, and order polytopes. Among other applications, we generalize a positivity statement from Schubert calculus by Fomin and Kirillov (1997) from straight shapes to skew shapes. We show that all shard polytopes are Ehrhart positive and, stronger, that all fence posets, including the zig-zag poset, and all circular fence posets have order polynomials with nonnegative coefficients. We discuss a general method for proving positivity which reduces to showing positivity of the linear terms of the order polynomials. We propose positivity conjectures on other relevant classes of posets.

Figures (10)

• Figure 1: The skew shape $\lambda/\mu=6533/21$, its cell poset $P_{\lambda/\mu}$, and a plane partition with entries in $\{0,\ldots,5\}$.
• Figure 2: Example of a fence poset.
• Figure 3: One minimum element, covered by $4$ elements.
• Figure 4: The Hasse diagram of the zig-zag poset $Z_9$.
• Figure 5: The diagram of a dominant permutation $w_{311}=4231$ (left), and the diagram and associated skew shape of the vexillary permutations $w=461532$ (center) and of $1^2\times w_{311}=126453$ (right).

Theorems & Definitions (51)

• Theorem 1.1: Main Theorem \ref{['thm:positivity_skew']}
• Corollary 1.2: Corollary \ref{['coro:fences-order-positive']}
• Corollary 1.3: Corollary \ref{['coro:shards']}
• Corollary 1.4
• Conjecture 1.5
• Theorem 1.6: Theorem \ref{['thm:circular-fences-order-positive']}
• Example 2.1
• Proposition 2.2: Kreweras Kreweras_1965
• Proposition 3.1
• proof