An arXiv preprint reports an exact way to calculate the facet gap between the chain and order polytopes of any finite poset. The gap is a difference in facet counts—the number of boundary pieces of the chain polytope minus the number of boundary pieces of the order polytope. The paper says this global difference is exactly the sum of local weights attached to star elements.
The paper’s central move is to replace the full count with an accounting rule for individual elements. For an element q, it defines the weight w(q) = (d(q) − 1)(r(q) − 1). An element is called a star when d(q) is at least 2 and r(q) is at least 2, and w(q) is greater than 0 precisely for star elements.
Written formally, the result says that for every poset P, the facet gap equals the sum of (d(q) − 1)(r(q) − 1) over q in P, or, equivalently, the sum of w(q) over the star elements. This is a deterministic mathematical identity rather than an estimate drawn from a sample.
A global comparison built from local pieces
The ordinary facet-gap result comes from a direct argument that does not use marked posets. The paper also gives formulas for both facet counts, then relates their difference to the local weights. That makes the calculation depend on the structure of the finite poset under consideration, not on statistical inference.
The local formula leads to an exact classification of a zero gap. The facet gap is 0 if and only if the poset has no star elements. In the paper’s terminology, the absence of stars is therefore precisely the condition for the two facet counts to match.
A gap of exactly 1 is also narrowly defined. It occurs if and only if there is exactly one star element, and that element has type (d(q), r(q)) = (2, 2).
For a gap of exactly 2, the paper identifies exactly two possibilities. One is exactly two simple star elements, both of type (2, 2). The other is exactly one star element of type (2, 3) or (3, 2). These are structural classifications of finite posets, not frequencies observed in an experimental sample.
The same local weight sum also supplies the requested upper bound, with equality. The result assigns that sum a value of 0 on X-avoiding posets and a value of 1 on X-orchids.
The same weights in a marked construction
The paper carries the weight idea into marked chain–order polytopes. This part concerns a regular marked poset with an admissible decomposition in which the chain part is an order ideal. The facet comparison is then made between admissible decompositions within that stated construction.
To compare those decompositions, the proof moves order-ideal elements into the chain part one at a time, following a poset-compatible order. A move involving a non-star element leaves the consecutive polytopes unimodularly equivalent, so it contributes no facet difference. A star move has an exact local facet increment equal to w(q).
Adding the stepwise changes gives the marked result: the facet-number difference between two admissible decompositions is obtained by subtracting their corresponding sums of star weights. In this family, the local weights determine the difference between the global facet counts.
A result with a defined boundary
The marked extension is not a claim about every order–chain construction. The paper says it does not apply to the Hibi–Li–Li–Mu–Tsuchiya order–chain polytopes because those use an edge decomposition rather than the vertex decomposition used here. No crossing-number bound is claimed for that family.
Whether analogous facet-gap or crossing-number results hold for that other family is left unaddressed. More broadly, the work provides mathematical results for finite posets and regular marked posets under the stated definitions; it reports no empirical, human or laboratory evidence.
The document header identifies the work as arXiv:2608.19989v1, dated 20 Aug 2026. The supplied publication record classifies it as a preprint, and the analysis reports no statistical uncertainty or real-world validation.
The author reports financial support from the Deutsche Forschungsgemeinschaft through Symbolic Tools in Mathematics and their Application, TRR 195, project-ID 286237555.
Paper data and sources
Original title: On facet gaps of order and chain polytopes
Authors: Ghislain Fourier
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text