Skip to main content

Partial Rank Symmetry of Distributive Lattices for Fences

Abstract

Associated with any composition \(\beta =(a,b,\ldots )\) is a corresponding fence poset \(F(\beta )\) whose covering relations are

$$\begin{aligned} x_1\lhd x_2 \lhd \ldots \lhd x_{a+1}\rhd x_{a+2}\rhd \ldots \rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots . \end{aligned}$$

The distributive lattice \(L(\beta )\) of all lower order ideals of \(F(\beta )\) is important in the theory of cluster algebras. In addition, its rank generating function \(r(q;\beta )\) is used to define q-analogues of rational numbers. Kantarcı Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth, and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that \(r(q;\beta )\) is unimodal. We show that, when \(\beta \) has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of \(F(\beta )\) of size k equals the number of filters of size k,  when k is below a certain value. Our proof is completely bijective. Kantarcı Oğuz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result, as well. We end with some questions and conjectures raised by this work.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Christos A. Athanasiadis. Edgewise subdivisions, local \(h\)-polynomials, and excedances in the wreath product \({\mathbb{Z}}_r \wr {\mathfrak{S}}_n\). SIAM J. Discrete Math., 28(3):1479–1492, 2014.

  2. Christos A. Athanasiadis. Gamma-positivity in combinatorics and geometry. Sém. Lothar. Combin., 77:Art. B77i, 64, [2016-2018].

  3. Matthias Beck, Katharina Jochemko, and Emily McCullough. \(h^\ast \)-polynomials of zonotopes. Trans. Amer. Math. Soc., 371(3):2021–2042, 2019.

    MathSciNet  Article  Google Scholar 

  4. Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. In Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), pages 437–483. CRC Press, Boca Raton, FL, 2015.

  5. Francesco Brenti. Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. In Jerusalem combinatorics ’93, volume 178 of Contemp. Math., pages 71–89. Amer. Math. Soc., Providence, RI, 1994.

  6. Petter Brändén and Liam Solus. Symmetric decompositions and real-rootedness. Int. Math. Res. Not. IMRN, (10):7764–7798, 2021.

    MathSciNet  Article  Google Scholar 

  7. Andrew Claussen. Expansion posets for polygon cluster algebras. PhD thesis, Michigan State University, 2020.

  8. Sergi Elizalde, Matthew Plante, Tom Roby, and Bruce E. Sagan. Rowmotion on fences, 2021. Preprint arXiv:math.CO/2108.12443.

  9. Curtis Greene and Daniel J. Kleitman. Strong versions of Sperner’s theorem. J. Combinatorial Theory Ser. A, 20(1):80–88, 1976.

    MathSciNet  Article  Google Scholar 

  10. Curtis Greene and Daniel J. Kleitman. Proof techniques in the theory of finite sets. In Studies in combinatorics, volume 17 of MAA Stud. Math., pages 22–79. Math. Assoc. America, Washington, D.C., 1978.

  11. Ezgi Kantarcı Oğuz and Mohan Ravichandran. Rank polynomials of fence posets are unimodal, 2021. Preprint arXiv:math.CO/2112.00518.

  12. Sophie Morier-Genoud and Valentin Ovsienko. \(q\)-deformed rationals and \(q\)-continued fractions. Forum Math. Sigma, 8:Paper No. e13, 55, 2020.

  13. Thomas McConville, Bruce E. Sagan, and Clifford Smyth. On a rank-unimodality conjecture of Morier-Genoud and Ovsienko. Discrete Math., 344(8):Paper No. 112483, 13, 2021.

  14. James Propp. The combinatorics of frieze patterns and Markoff numbers. Integers, 20:Paper No. A12, 38, 2020.

  15. Tom Roby. Dynamical algebraic combinatorics and the homomesy phenomenon. In Recent trends in combinatorics, volume 159 of IMA Vol. Math. Appl., pages 619–652. Springer, [Cham], 2016.

  16. Bruce E. Sagan. Combinatorics: the art of counting, volume 210 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2020] 2020.

  17. Ralf Schiffler. A cluster expansion formula (\(A_n\) case). Electron. J. Combin., 15(1):Research paper 64, 9, 2008.

  18. Ralf Schiffler. On cluster algebras arising from unpunctured surfaces. II. Adv. Math., 223(6):1885–1923, 2010.

    MathSciNet  Article  Google Scholar 

  19. Liam Solus. Simplices for numeral systems. Trans. Amer. Math. Soc., 371(3):2089–2107, 2019.

    MathSciNet  Article  Google Scholar 

  20. Ralf Schiffler and Hugh Thomas. On cluster algebras arising from unpunctured surfaces. Int. Math. Res. Not. IMRN, (17):3160–3189, 2009.

    MathSciNet  MATH  Google Scholar 

  21. Richard P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph theory and its applications: East and West (Jinan, 1986), volume 576 of Ann. New York Acad. Sci., pages 500–535. New York Acad. Sci., New York, 1989.

  22. Richard P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012.

  23. Jessica Striker. Dynamical algebraic combinatorics: promotion, rowmotion, and resonance. Notices Amer. Math. Soc., 64(6):543–549, 2017.

    MathSciNet  Article  Google Scholar 

  24. Jan Schepers and Leen Van Langenhoven. Unimodality questions for integrally closed lattice polytopes. Ann. Comb., 17(3):571–589, 2013.

    MathSciNet  Article  Google Scholar 

  25. Toshiya Yurikusa. Cluster expansion formulas in type A. Algebr. Represent. Theory, 22(1):1–19, 2019.

    MathSciNet  Article  Google Scholar 

  26. Toshiya Yurikusa. Combinatorial cluster expansion formulas from triangulated surfaces. Electron. J. Combin., 26(2):Paper No. 2.33, 39, 2019.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bruce E. Sagan.

Ethics declarations

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by Vasu Tewari.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Elizalde, S., Sagan, B.E. Partial Rank Symmetry of Distributive Lattices for Fences. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00600-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00026-022-00600-8

Keywords

  • Bottom heavy
  • Bottom interlacing
  • Distributive lattice
  • Fence
  • Gate
  • Order ideal
  • Poset
  • Rank
  • Symmetric
  • Unimodal
  • Top heavy
  • Top interlacing

Mathematics Subject Classification

  • Primary 06A07
  • Secondary 05A15
  • 05A19
  • 05A20
  • 06D05