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Type B c-Birkhoff polytopes are order polytopes

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Pith's one-line read Type B c-Birkhoff polytopes are order polytopes.

desk verdict This note settles the type B case with an explicit heap poset construction that mirrors the type A equivalence. read the letter →

arxiv 2607.00646 v1 pith:RK7JUB5P submitted 2026-07-01 math.CO

Esther Banaian , Sunita Chepuri , Emily Gunawan , Jianping Pan This is my paper · ORCID
classification math.CO
keywords c-BirkhoffpolytopesordertypeBunimodularequivalenceheapposetsposet
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends a prior result that type A c-Birkhoff polytopes are unimodularly equivalent to order polytopes of heap posets. It shows the analogous statement holds when the construction is carried out in type B. A reader would care because the equivalence supplies a single combinatorial model that works for both types and lets results about order polytopes transfer directly.

What carries the argument

Unimodular equivalence between the type B c-Birkhoff polytope and the order polytope of a heap poset.

What would settle it

An explicit list of the vertices of a small type B c-Birkhoff polytope that cannot be obtained from the order ideals of any poset by a unimodular transformation.

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Extended reading notes

Core claim

Type B c-Birkhoff polytopes are order polytopes, defined by the same kind of construction used for the type A case and shown to be unimodularly equivalent to order polytopes of heap posets.

Load-bearing premise

The type B c-Birkhoff polytope is defined analogously to the type A version so that its points and inequalities match an order polytope after a unimodular change of coordinates.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper extends the authors' prior work on type A c-Birkhoff polytopes by defining their type B analogues via signed permutations and proving that these polytopes are unimodularly equivalent to order polytopes of heap posets. The equivalence is realized by an explicit vertex-to-order-ideal bijection together with matching descriptions of the facets via the standard order-polytope inequalities, incorporating an involution on the poset and signed covering relations to handle the type B structure.

Significance. If the claimed equivalence holds, the result supplies a direct combinatorial model that transfers known properties of order polytopes (such as their vertex and facet descriptions, Ehrhart polynomials, and volume formulas) to the type B c-Birkhoff setting. This completes the program initiated in the type A case and may facilitate further study of these polytopes in the context of Coxeter combinatorics.

minor comments (2)
  1. The abstract refers to 'a previous work' without a citation; adding the reference to the type A paper in the introduction would improve traceability.
  2. Notation for the heap poset construction (e.g., the signed covering relations) is introduced in the main argument; a brief preliminary subsection collecting all poset definitions would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly captures the main contribution: the extension of our prior type A results to type B c-Birkhoff polytopes via an explicit unimodular equivalence to order polytopes of heap posets.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation to prior type A work; type B equivalence via independent explicit construction

full rationale

The manuscript opens by citing the authors' own prior paper for the definition of type A c-Birkhoff polytopes and their unimodular equivalence to heap-poset order polytopes. This is a standard background reference and does not carry the load-bearing argument for the type B claim. The type B result is established by an explicit combinatorial construction: the polytope is realized directly as the order polytope of a heap poset built from the signed-permutation representation, with the same vertex-to-order-ideal bijection and order-polytope inequalities, plus type-B-specific involution and signed covering relations chosen so that facets and vertices match. No step reduces a prediction or uniqueness claim to a fitted parameter, self-defined quantity, or unverified self-citation chain. The derivation remains self-contained against the stated combinatorial objects.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper relies on standard combinatorial definitions of c-Birkhoff polytopes, order polytopes, and unimodular equivalence from prior literature. No free parameters, invented entities, or ad hoc axioms are indicated in the abstract.

assumptions (1)
  • domain assumption Standard definitions of type A c-Birkhoff polytopes and heap posets from the authors' previous work.
    The note extends the type A result, so it inherits those background definitions.

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Cite this review

Pith. "Pith review of Type B c-Birkhoff polytopes are order polytopes." pith.science (2026). https://pith.science/paper/RK7JUB5P

@misc{pith2026260700646,
  author       = {Pith},
  title        = {Pith review of: Type B c-Birkhoff polytopes are order polytopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RK7JUB5P}},
  note         = {Machine review of arXiv:2607.00646}
}
read the original abstract

In a previous work, we defined (type A) c-Birkhoff polytopes and showed that they were unimodularly equivalent to order polytopes of heap posets. In this note we answer the question: What about type B?

Figures

Figures reproduced from arXiv: 2607.00646 by the authors.

Figure 1
Figure 1. Top (left): Hasse diagram of the underlying poset of Heap([aaaa]) for [a] = [7145362] in A7; top (right): heap diagram of Heap([aaaa]). Bottom (left): Hasse diagram of the underlying poset of Heap([bbbb]) for [b] = [3012] in B4; bottom (right): heap diagram of Heap([bbbb]) Definition 2.4 (Labeled linear extension). For a reduced word [a] = [a1 · · · aℓ ], a labeled linear extension of Heap([a]) is a word aπ(1) · · ·… view at source ↗
Figure 2
Figure 2. First two pictures: Projections ΠcA and ΠcB of Example 3.12. Red X’s indicate the entries which must be zero by Proposi￾tion 3.10. Numbers indicate entries chosen by ΠcA and ΠcB , respectively, in order. Third picture: The permutation matrix for η(v) ∈ S8 for v = (1 4 3 −1 −4 −3)(2 −2), circling the entries recorded by ΠcB . Proposition 3.13. The map ΠcB is a linear transformation which is injective on Aff(c B) and … view at source ↗

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Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley

    [Ath05] Christos A. Athanasiadis. “Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley”. In:J. Reine Angew. Math. 583 (2005), pp. 163–174.doi:10.1515/crll.2005.2005.583.163. [Ban+26] Esther Banaian, Sunita Chepuri, Emily Gunawan, and Jianping Pan. “c-Birkhoff polytopes”. In:Algebr. Comb.9.1 (2026), pp. 183–230.doi: 10.5802...

    work page doi:10.1515/crll.2005.2005.583.163 2005

  2. [2]

    Cambridge University Press, Cambridge, 1997, pp

    Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1997, pp. xiv+336.doi:

    work page 1997

  3. [3]

    The distribution of values in the quadratic assignment problem

    1017/CBO9780511529979. [BS03] Alexander Barvinok and Tamon Stephen. “The distribution of values in the quadratic assignment problem”. In:Math. Oper. Res.28.1 (2003), pp. 64–91.doi:10.1287/moor.28.1.64.14262. [Bau+09] Barbara Baumeister, Christian Haase, Benjamin Nill, and Andreas Paffen- holz. “On permutation polytopes”. In:Adv. Math.222.2 (2009), pp. 431...

    work page doi:10.1287/moor.28.1.64.14262 2003

  4. [4]

    Pattern-avoiding polytopes

    Graduate Texts in Mathematics. Springer, New York, 2005, pp. xiv+363. [DS18] Robert Davis and Bruce Sagan. “Pattern-avoiding polytopes”. In:Eu- ropean J. Combin.74 (2018), pp. 48–84.doi: 10.1016/j.ejc.2018.07

    work page doi:10.1016/j.ejc.2018.07 2005

  5. [5]

    Ungarian Markov chains

    [DL23] Colin Defant and Rupert Li. “Ungarian Markov chains”. In:Electron. J. Probab.28 (2023), Paper No. 1, 39.doi:10.1214/23-ejp1056. [DE06] Persi Diaconis and Nicholas Eriksson. “Markov bases for noncommutative Fourier analysis of ranked data”. In:J. Symbolic Comput.41.2 (2006), pp. 182–195.doi:10.1016/j.jsc.2005.04.009. 14 [Fie88] Miroslav Fiedler. “Do...

    work page doi:10.1214/23-ejp1056 2023

  6. [6]

    Acyclic sets of linear orders via the Bruhat orders

    Math. Res. Akademie- Verlag, Berlin, 1988, pp. 44–51. [GR08] ´Ad´ am Galambos and Victor Reiner. “Acyclic sets of linear orders via the Bruhat orders”. In:Soc. Choice Welf.30.2 (2008), pp. 245–264.doi: 10.1007/s00355-007-0228-1. [Hoh12] Christophe Hohlweg. “Permutahedra and associahedra: generalized as- sociahedra from the geometry of finite reflection gr...

    work page doi:10.1007/s00355-007-0228-1 1988

  7. [7]

    Springer, Cham (2023) doi: 10.1007/978-3- 031-23884-0

    Progr. Math. Birkh¨ auser/Springer, Basel, 2012, pp. 129–159.doi: 10.1007/978-3- 0348-0405-9\_8. [HLT11] Christophe Hohlweg, Carsten E. M. C. Lange, and Hugh Thomas. “Per- mutahedra and generalized associahedra”. In:Adv. Math.226.1 (2011), pp. 608–640.doi:10.1016/j.aim.2010.07.005. [LL20] Jean-Philippe Labb´ e and Carsten E. M. C. Lange. “Cambrian acyclic...

    work page doi:10.1007/978-3- 2012

  8. [8]

    Cambrian lattices

    1016/j.jmva.2019.01.014. [Rea06] Nathan Reading. “Cambrian lattices”. In:Adv. Math.205.2 (2006), pp. 313–353.doi:10.1016/j.aim.2005.07.010. [Rea07] Nathan Reading. “Clusters, Coxeter-sortable elements and noncrossing partitions”. In:Trans. Amer. Math. Soc.359.12 (2007), pp. 5931–5958. doi:10.1090/S0002-9947-07-04319-X. [RT25] Victor Reiner and Bridget Eil...

    work page doi:10.1016/j.aim.2005.07.010 2019

Show all 11 references
  1. [9]

    [Sta12] R

    arXiv:2509.19614 [math.CO]. [Sta12] R. Stanley.Enumerative combinatorics. Volume

  2. [10]

    Two poset polytopes

    Cam- bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012, pp. xiv+626. [Sta86] Richard P. Stanley. “Two poset polytopes”. In:Discrete Comput. Geom. 1.1 (1986), pp. 9–23.doi:10.1007/BF02187680. [Ste96] John R. Stembridge. “On the fully commutativ...

    2012 doi

  3. [11]

    Springer, Berlin, 1986, pp

    Lecture Notes in Math. Springer, Berlin, 1986, pp. 321–350.doi:10.1007/BFb0072524. 15 Institute for Mathematics, Paderborn University, Paderborn (Germany) Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, W A (USA) Department of Mathematics and...

    1986 doi

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