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On Nested Chain Decompositions of Normalized Matching Posets of Rank 3

An Erratum to this article was published on 21 August 2010

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.

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Correspondence to Shahriar Shahriari.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s11083-010-9173-1

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Escamilla, E.G., Nicolae, A.C., Salerno, P.R. et al. On Nested Chain Decompositions of Normalized Matching Posets of Rank 3. Order 28, 357–373 (2011). https://doi.org/10.1007/s11083-010-9164-2

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  • DOI: https://doi.org/10.1007/s11083-010-9164-2

Keywords

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

Mathematics Subject Classifications (2010)

  • Primary 06A07; Secondary 05D05
  • 05D15