Skip to main content
SpringerLink
Log in

Level Eulerian Posets

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem–Mahler–Lech theorem, the ab-series of a level poset is shown to be a rational generating function in the non-commutative variables a and b. In the case the poset is also Eulerian, the analogous result holds for the cd-series. Using coalgebraic techniques a method is developed to recognize the cd-series matrix of a level Eulerian poset.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babson E., Hersh P.: Discrete Morse functions from lexicographic orders. Trans. Am. Math. Soc. 357, 509–534 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bayer M., Billera L.J.: Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79, 143–157 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bayer M., Klapper A.: A new index for polytopes. Discret. Comput. Geom. 6, 33–47 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bayer M., Hetyei G.: Flag vectors of Eulerian partially ordered sets. Eur. J. Combin. 22, 5–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bayer M., Hetyei G.: Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function. Discret. Math. 256, 577–593 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Billera L.J., Hetyei G.: Linear inequalities for flags in graded partially ordered sets. J. Combin. Theory Ser. A 89, 77–104 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Billera L.J., Hetyei G.: Decompositions of partially ordered sets. Order 17, 141–166 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Björner A.: Topological methods in Handbook of combinatorics, vols. 1, 2., pp. 1819. Elsevier, Amsterdam (1995)

    Google Scholar 

  9. Björner A., Wachs M.: Bruhat order of Coxeter groups and shellability. Adv. Math. 43, 87–100 (1982)

    Article  MATH  Google Scholar 

  10. Björner A., Wachs M.: On lexicographically shellable posets. Trans. Am. Math. Soc. 277, 323–341 (1983)

    Article  MATH  Google Scholar 

  11. Ehrenborg R.: k-Eulerian posets. Order 18, 227–236 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ehrenborg R., Readdy M.: Coproducts and the cd-index J. Algebr. Combin. 8, 273–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ehrenborg R., Readdy M.: Homology of Newtonian coalgebras. Eur. J. Combin. 23, 919–927 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ehrenborg R., Readdy M.: Classification of the factorial functions of Eulerian binomial and Sheffer posets. J. Combin. Theory Ser. A 114, 339–359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Forman R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gross J., Tucker T.W. (2001) Topological graph theory. Reprint of the 1987 original (Wiley, New York) with a new preface and supplementary bibliography. Dover Publications, Inc., Mineola

  17. Heap B.R., Lynn M.S.: The structure of powers of nonnegative matrices. I. The index of convergence. SIAM J. Appl. Math. 14, 610–639 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  18. Holladay J.C., Varga R.S.: On powers of non-negative matrices. Proc. Am. Math. Soc. 9, 631–634 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kozlov D.: General lexicographic shellability and orbit arrangements. Ann. Combin. 1, 67–90 (1997)

    Article  MATH  Google Scholar 

  20. Lech C.: A note on recurring series. Ark. Mat. 2, 417–421 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mahler K.: On the Taylor coefficients of rational functions. Proc. Camb. Philos. Soc. 52, 39–48 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  22. Perrin, D., Schützenberger, M-P.: Synchronizing prefix codes and automata and the road coloring problem, In: Symbolic dynamics and its applications (New Haven, CT, 1991). Contemp. Math., 135, 295–318. Am. Math. Soc., Providence (1992)

  23. Pták V.: On a combinatorial theorem and its application to nonnegative matrices. Czechoslov. Math. J. 8(83), 487–495 (1958)

    Google Scholar 

  24. Pták V., Sedlaček I.: On the index of imprimitivity of nonnegative matrices. Czechoslov. Math. J. 8(83), 496–501 (1958)

    Google Scholar 

  25. Rosenblatt D.: On the graphs and asymptotic forms of finite Boolean relation matrices and stochastic matrices. Naval. Res. Logist. Quart. 4, 151–167 (1957)

    Article  MathSciNet  Google Scholar 

  26. Sachkov, V.N., Tarakanov, V.E.: Combinatorics of nonnegative matrices. Translations of Mathematical Monographs, vol. 213. The American Mathematical Society, Providence (2002)

  27. Skolem T.: Einige Sätze über gewisse Reihenentwicklungen und exponentiale Beziehungen mit Anwendung auf diophantische Gleichungen. Oslo Vid. Akad. Skrifter I 6, 1–61 (1933)

    Google Scholar 

  28. Stanley R.P.: Flag f-vectors and the cd-index. Math. Z. 216, 483–499 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stanley R.P.: Enumerative Combinatorics, vol. I. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  30. Stanley R.P.: Enumerative Combinatorics, vol. II. . Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  31. Wachs, M.: Poset topology: tools and applications. In: Miller, E., Reiner, V., Sturmfels, B. (eds.) Geometric combinatorics. IAS/Park City Math. Series, 13, pp. 497–615. Am. Math. Soc., Providence (2007)

Download references

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 Richard Ehrenborg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ehrenborg, R., Hetyei, G. & Readdy, M. Level Eulerian Posets. Graphs and Combinatorics 29, 857–882 (2013). https://doi.org/10.1007/s00373-012-1173-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-012-1173-z

Keywords

Mathematics Subject Classification

Search

Navigation