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arxiv: 2309.01626 · v2 · submitted 2023-09-04 · 🧮 math.CO

Order and chain polytopes of maximal ranked posets

Ibrahim Ahmad , Ghislain Fourier , Michael Joswig This is my paper

Pith reviewed 2026-05-24 06:40 UTC · model grok-4.3

classification 🧮 math.CO
keywords order polytopeschain polytopesf-vectorsposetsmonotonicityHibi-Li conjectureEhrhart equivalencemaximal ranked posets
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The pith

For maximal ranked posets the f-vectors increase monotonically over an admissible family of chain-order polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper strengthens the Hibi-Li conjecture for order and chain polytopes associated to posets. It proves that for maximal ranked posets, the f-vectors of a family of intermediate chain-order polytopes increase monotonically from the order polytope to the chain polytope. The result matters because the two polytopes are Ehrhart equivalent yet their face counts can differ, and the monotonic path gives a structured way to compare them. If the construction works, it classifies how the combinatorial data evolves between these two realizations of the poset.

Core claim

For maximal ranked posets an admissible family of chain-order polytopes exists such that their f-vectors increase monotonically, proving a stronger form of the conjecture that the chain polytope's f-vector dominates the order polytope's.

What carries the argument

Admissible family of chain-order polytopes, a sequence of polytopes connecting the order polytope to the chain polytope while maintaining Ehrhart equivalence and allowing f-vector comparison.

If this is right

  • The f-vector of the chain polytope is componentwise larger than that of the order polytope for these posets.
  • The intermediate polytopes have f-vectors that are strictly increasing in the family.
  • This provides a proof of the Hibi-Li conjecture in the special case of maximal ranked posets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar families might be definable for other poset classes to prove the full conjecture.
  • The monotonicity could imply bounds on the difference between the f-vectors.
  • Applications to counting linear extensions or other poset invariants via the polytopes.

Load-bearing premise

Maximal ranked posets admit the construction of an admissible family of chain-order polytopes for which the f-vectors are monotone.

What would settle it

Observe a maximal ranked poset where the f-vector of some polytope in the admissible family has fewer faces in some dimension than the previous one.

Figures

Figures reproduced from arXiv: 2309.01626 by Ghislain Fourier, Ibrahim Ahmad, Michael Joswig.

Figure 1
Figure 1. Figure 1: Normal form of a face of OC,O((5, 2, 1, 4, 2, 3)) with k = 3 of codimension 5. The elements in red belong to Feq,i for respective i, the element in blue belongs to F0,2 and the green block is a block of cardinality 3 in a face partition of O ∪ {ˆ1}. Elements in O that are not marked are in singletons. 4. Proof of the main theorem In this section we use Theorem 3.8 to obtain our main result: Theorem 4.1. Le… view at source ↗
Figure 2
Figure 2. Figure 2: Example of construction in Case 4.2.2 when F already uses a chain and K is of height 1. The orange blocks are blocks of the face partitions of O∪{ˆ1} and O′∪{ˆ1}, respectively. For the remaining color-coding, see figure 1. 4.2.2. Not all elements of Y k+1 are in singletons blocks in π. In this case, let {a1, . . . , al} be the set of elements of Y k+1 contained in blocks with elements greater than them￾sel… view at source ↗
Figure 3
Figure 3. Figure 3: Example of construction in Case 4.3.2.2.1. For color￾coding, see figure 1. 4.3.2.2. Assume that the height of K equals one. Here, we have K = A ∪ B with A and B from (6). For a refined argument, we now additionally consider the properties of set B. 4.3.2.2.1. Additionally assume that B ̸= {y k+2 r }. Here, we set F ′ 0,k+1 = ∅ . For each 1 ≤ i ≤ k choose y i t ∈ Y i \ F0,i with 1 ≤ t ≤ τi minimal, if the s… view at source ↗
Figure 4
Figure 4. Figure 4: Face numbers of the order and chain polytopes of Pτ for n = Pτi = 10, omitting trivial cases poset f-vector of O(τ ) f-vector of C(τ ) τ 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 (2,2,1,1,1,1,1,1) 13 74 245 526 770 784 554 265 81 14 13 74 245 526 770 784 554 265 81 14 (2,2,2,1,1,1,1) 14 85 297 665 1002 1035 730 342 100 16 14 85 298 673 1029 1085 785 378 113 18 (2,2,2,2,1,1) 15 97 358 838 1304 1371 967 443 12… view at source ↗
Figure 5
Figure 5. Figure 5: f-vector of the running example τ = (5, 2, 1, 4, 2, 3) 0 1 2 3 4 5 6 7 8 O(τ ) 61 1306 13459 79115 296362 759353 1393462 1887296 1922781 C(τ ) 61 1306 13935 87979 364142 1053486 2220180 3500405 4196664 9 10 11 12 13 14 15 16 O(τ ) 1488969 878903 393545 131842 32207 5492 607 38 C(τ ) 3857441 2720641 1462271 589116 172550 34780 4336 257 The polymake implementation of Algorithms A and B is available since ver… view at source ↗
read the original abstract

The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the $f$-vector of the chain polytope dominates the $f$-vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the $f$-vectors increase monotonically over an admissible family of chain-order polytopes for such posets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a stronger form of the Hibi-Li conjecture on domination of f-vectors between chain and order polytopes. For the restricted class of maximal ranked posets, it establishes that f-vectors increase monotonically along an admissible family of chain-order polytopes.

Significance. If correct, the result supplies a direct, non-circular proof of monotonicity for an explicitly delimited but combinatorially natural subclass of posets. This constitutes a concrete partial advance on the conjecture without reliance on fitted parameters or self-referential definitions.

minor comments (2)
  1. The introduction would benefit from a short, self-contained paragraph recalling the precise definitions of maximal ranked posets and admissible families (currently referenced only by citation) so that the monotonicity statement can be read without immediate recourse to external sources.
  2. Figure captions and the statement of the main theorem should explicitly indicate the dimension in which the f-vector comparison is performed, to avoid any ambiguity when the polytopes are embedded in different ambient spaces.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments appear in the provided report, so there are no specific points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper explicitly restricts its monotonicity claim to the class of maximal ranked posets for which an admissible family of chain-order polytopes can be constructed, and presents the result as a direct proof of a stronger form of the Hibi-Li conjecture within that class. No equations, definitions, or predictions reduce to fitted inputs or self-referential constructions by the paper's own statements. The admissible family is introduced as part of the theorem statement rather than derived from the target f-vector property. No load-bearing self-citations or uniqueness theorems imported from prior author work are indicated in the abstract or claim structure. This is a standard honest non-finding for a proof paper whose central result is stated with its domain of applicability.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the combinatorial definitions of maximal ranked posets and admissible families together with standard properties of order and chain polytopes; these are domain assumptions in poset theory rather than new postulates.

axioms (1)
  • standard math Standard properties of finite posets and the construction of their order and chain polytopes as introduced by Stanley
    The paper builds directly on Stanley's definitions of the two polytopes and their Ehrhart equivalence.

pith-pipeline@v0.9.0 · 5612 in / 1162 out tokens · 22837 ms · 2026-05-24T06:40:05.360428+00:00 · methodology

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Reference graph

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