Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We show that the Simion type B associahedron is combinatorially equivalent to a pulling triangulation of the type A root polytope known as the Legendre polytope. Furthermore, we show that every pulling triangulation of the boundary of the Legendre polytope yields a flag complex. Our triangulation refines a decomposition of the boundary of the Legendre polytope given by Cho. Finally, we extend Cho’s cyclic group action to the triangulation in such a way that it corresponds to rotating centrally symmetric triangulations of a regular \((2n+2)\)-gon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Ardila, F., Beck, M., Hoşten, S., Pfeifle, J., Seashore, K.: Root polytopes and growth series of root lattices. SIAM J. Discrete Math. 25(1), 360–378 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Athanasiadis, C.A.: \(h^*\)-Vectors, Eulerian polynomials and stable polytopes of graphs. Electron. J. Comb. 11(2), R6 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Billera, L.J., Cushman, R., Sanders, J.A.: The Stanley decomposition of the harmonic oscillator. Nederl. Acad. Wetensch. Indag. Math. 50(4), 375–393 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bott, R., Taubes, C.: On the self-linking of knots. Topology and physics. J. Math. Phys. 35(10), 5247–5287 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burgiel, H., Reiner, V.: Two signed associahedra. N. Y. J. Math. 4, 83–95 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Ceballos, C., Padrol, A., Sarmiento, C.: Geometry of \(\nu \)-Tamari lattices in types \(A\) and \(B\). Sémin. Lothar. Combin. 78, B68 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Ceballos, C., Santos, F., Ziegler, G.M.: Many non-equivalent realizations of the associahedron. Combinatorica 35(5), 513–551 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cellini, P., Marietti, M.: Root polytopes and Abelian ideals. J. Algebr. Comb. 39(3), 607–645 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cho, S.: Polytopes of roots of type \(A_n\). Bull. Aust. Math. Soc. 59(3), 391–402 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clark, E., Ehrenborg, R.: Excedances of affine permutations. Adv. Appl. Math. 46(1–4), 175–191 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cori, R., Hetyei, G.: Counting genus one partitions and permutations. Sémin. Lothar. Comb. 70, B70e (2013)

    MathSciNet  MATH  Google Scholar 

  12. Gelfand, I.M., Graev, M.I., Postnikov, A.: Combinatorics of hypergeometric functions associated with positive roots. In: Arnold–Gelfand Mathematical Seminars: Geometry and Singularity Theory, pp. 205–221. Birkhäuser, Boston (1996)

  13. Hetyei, G.: Delannoy orthants of Legendre polytopes. Discrete Comput. Geom. 42(4), 705–721 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hudson, J.F.P.: Piecewise Linear Topology. University of Chicago Lecture Notes prepared with the assistance of J.L. Shaneson and J. Lees. W.A. Benjamin, New York (1969)

    Google Scholar 

  15. Lee, C.W.: The associahedron triangulations of the \(n\)-gon. Eur. J. Comb. 10(6), 551–560 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. MacMahon, P.A.: Combinatory Analysis, vol. I, II (bound in one volume). Reprint of An Introduction to Combinatory Analysis (1920) and Combinatory Analysis, vol. I, II (1915, 1916). Dover Phoenix Editions. Dover Publications, Inc., Mineola, NY (2004)

  17. Markl, M.: Simplex, associahedron, and cyclohedron. In: McCleary, J. (ed.) Higher Homotopy Structures in Topology and Mathematical Physics. Contemporary Mathematics, vol. 227, pp. 235–265. American Mathematical Society, Providence, RI (1999)

    Chapter  Google Scholar 

  18. Mészáros, K.: Root polytopes, triangulations, and the subdivision algebra. I. Trans. Am. Math. Soc. 363(8), 4359–4382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mészáros, K.: Root polytopes, triangulations, and the subdivision algebra. II. Trans. Am. Math. Soc. 363(11), 6111–6141 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Poincaré, H.: Second complement a l’analysis situs. Proc. Lond. Math. Soc. 32, 277–308 (1900)

    Article  MathSciNet  MATH  Google Scholar 

  21. Postnikov, A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN 2009(6), 1026–1106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)

    MATH  Google Scholar 

  23. Simion, R.: A type-B associahedron. Adv. Appl. Math. 30(1–2), 2–25 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Veblen, O., Franklin, P.: On matrices whose elements are integers. Ann. Math. 23(1), 1–15 (1921)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the Princeton University Mathematics Department where this research was initiated. The first author was partially funded by the National Security Agency Grant H98230-13-1-028. This work was partially supported by grants from the Simons Foundation (# 429370 to Richard Ehrenborg, # 245153 and # 514648 to Gábor Hetyei, # 206001 and # 422467 to Margaret Readdy). The authors also thank the three anonymous referees for their comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Richard Ehrenborg & Margaret Readdy

  2. Department of Mathematics and Statistics, UNC-Charlotte, Charlotte, NC, 28223-0001, USA

    Gábor Hetyei

Authors
  1. Richard Ehrenborg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gábor Hetyei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Margaret Readdy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Margaret Readdy.

Additional information

Editor in Charge: János Pach

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ehrenborg, R., Hetyei, G. & Readdy, M. Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope. Discrete Comput Geom 60, 98–114 (2018). https://doi.org/10.1007/s00454-018-9973-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-018-9973-4

Keywords

Mathematics Subject Classification

Search

Navigation