Advertisement

Order

, Volume 28, Issue 2, pp 357–373 | Cite as

On Nested Chain Decompositions of Normalized Matching Posets of Rank 3

  • Elinor Gardner Escamilla
  • Andreea Cristina Nicolae
  • Paul Russell Salerno
  • Shahriar Shahriari
  • Jordan Olliver Tirrell
Article

Abstract

In 1975, J. Griggs conjectured that a normalized matching rank-unimodal poset possesses a nested chain decomposition. This elegant conjecture remains open even for posets of rank 3. Recently, Hsu, Logan, and Shahriari have made progress by developing techniques that produce nested chain decompositions for posets with certain rank numbers. As a demonstration of their methods, they prove that the conjecture is true for all rank 3 posets of width at most 7. In this paper, we present new general techniques for creating nested chain decompositions, and, as a corollary, we demonstrate the validity of the conjecture for all rank 3 posets of width at most 11.

Keywords

Chain decompositions Nested posets Griggs nesting conjecture Normalized matching property LYM property Saturated partition 

Mathematics Subject Classifications (2010)

Primary 06A07; Secondary 05D05 05D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, I.: Some problems in combinatorial number theory. Ph.D. thesis, University of Nottingham, Nottingham, United Kingdom (1967)Google Scholar
  2. 2.
    Anderson, I.: Combinatorics of Finite Sets. Dover Publications, Mineola, NY (2002). Corrected reprint of the 1989 edition published by Oxford University Press, OxfordzbMATHGoogle Scholar
  3. 3.
    Bollobás, B.: On generalized graphs. Acta Math. Acad. Sci. Hung. 16, 447–452 (1965)zbMATHCrossRefGoogle Scholar
  4. 4.
    Engel, K.: Sperner theory. In: Encyclopedia of Mathematics. Cambridge University Press, Cambridge (1997)Google Scholar
  5. 5.
    Gansner, E.R.: On the lattice of order ideals of an up-down poset. Discrete Math. 39, 113–122 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Graham, R.L., Harper, L.H.: Some results on matching in bipartite graphs. SIAM J. Appl. Math. 17, 1017–1022 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Greene, C., Kleitman, D.J.: The Structure of Sperner k-Families. J. Comb. Theory, Ser. A 20, 80–88 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Griggs, J.R.: Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math. 32, 807–809 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Griggs, J.R.: Symmetric Chain Orders, Sperner Theorems, and Loop Matchings. Ph.D. thesis, MIT (1977)Google Scholar
  10. 10.
    Griggs, J.R.: On chains and Sperner k-families in ranked posets. J. Comb. Theory, Ser. A 28, 156–168 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Griggs, J.R.: Problems on chain partitions. Discrete Math. 72, 157–162 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Griggs, J.R.: Matchings, cutsets, and chain partitions in graded posets. Discrete Math. 144, 33–46 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Harper, L.H.: The morphology of partially ordered sets. J. Comb. Theory, Ser. A 17, 44–58 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hsieh, W.N., Kleitman, D.J.: Normalized matching in direct products of partial orders. Stud. Appl. Math. 52, 285–289 (1973)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hsu, T., Logan, M., Shahriari, S.: Methods for nesting rank 3 normalized matching rank-unimodal posets. Discrete Math. 309(3), 521–531 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hsu, T., Logan, M., Shahriari, S., Towse, C.: Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size. Eur. J. Comb. 24, 219–228 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hsu, T., Logan, M., Shahriari, S.: The generalized Füredi conjecture holds for finite linear lattices. Discrete Math. 306(23), 3140–3144 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kleitman, D.J.: On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications. In: Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), Part 2: Graph Theory; Foundations, Partitions and Combinatorial Geometry, pp. 77–90. Math. Centrum, Amsterdam (1974). Math. Centre Tracts, No. 56Google Scholar
  19. 19.
    Lubell, D.: A short proof of Sperner’s lemma. J. Comb. Theory 1, 299 (1966)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mešalkin, L.D.: A generalization of Sperner’s theorem on the number of subsets of a finite set. Teor. Veroatn ee Primen. 8, 219–220 (1963)Google Scholar
  21. 21.
    Pearsall, A., Shahriari, S.: Chain decompositions of normalized matching posets of rank 2. To appear in the Lecture Notes Series of the Ramanujan Mathematical SocietyGoogle Scholar
  22. 22.
    Perfect, H.: Addendum to: “A short proof of the existence of k-saturated partitions of partially ordered sets” [Adv. in Math. 33(3), 207–211 (1979); MR0546293 (82c:06008)] by M. Saks, Glasgow Math. J. 25(1), 31–33 (1984)Google Scholar
  23. 23.
    Saks, M.: A short proof of the existence of k-saturated partitions of partially ordered sets. Adv. Math. 33(3), 207–211 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Shelley, K.: Matchwebs. Master’s thesis, San José State University (2007)Google Scholar
  25. 25.
    Wang, Y.: Nested chain partitions of LYM posets. Discrete Appl. Math. 145(3), 493–497 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    West, D.B., Harper, L.H., Daykin, D.E.: Some remarks on normalized matching. J. Comb. Theory, Ser. A 35, 301–308 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Yamamoto, K.: Logarithmic order of free distributive lattice. J. Math. Soc. Jpn. 6, 343–353 (1954)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Elinor Gardner Escamilla
    • 1
  • Andreea Cristina Nicolae
    • 1
  • Paul Russell Salerno
    • 1
  • Shahriar Shahriari
    • 1
  • Jordan Olliver Tirrell
    • 2
  1. 1.Department of MathematicsPomona CollegeClaremontUSA
  2. 2.Department of MathematicsLafayette CollegeEastonUSA

Personalised recommendations