Order

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# Some Remarks on Nestings in the Normalized Matching Posets of Rank 3

Article

## Abstract

In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset has a nested chain decomposition. This conjecture is proved to be true only for some posets of small ranks (Wang Discrete Math. 145(3), 493–497, 2005; Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011). In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfy the Griggs’s conjecture by refining the proofs in the two papers (Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011).

## Keywords

Nested chain decomposition Normalized matching posets Griggs nesting conjecture

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## References

1. 1.
Anderson, I.: Some Problems in Combinatorial Number Theory. Ph.D. Thesis University of Nottingham, Nottingham (1967)Google Scholar
2. 2.
Anderson, I.: Combinatorics of Finite Sets. Dover Publications, New York (2002)
3. 3.
Engel, K.: Sperner Theory. Cambridge University Press, New York (1997)
4. 4.
Escamilla, E.G., Nicolae, A.C., Salerno, P.R., Shahriari, S., Tirrell, J.O.: On nested chain decompositions of normalized matching posets of rank 3. Order 28, 357–373 (2011)
5. 5.
Griggs, J.R.: Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math. 32, 807–809 (1977)
6. 6.
Griggs, J.R.: On chains and Sperner k-families in ranked posets. J. Combin. Theory A 28, 156–168 (1980)
7. 7.
Griggs, J.R.: Problems on chain partitions. Discrete Math. 72, 157–162 (1988)
8. 8.
Hall, P.: On representative of subsets. J. London Math. Soc. 10, 26–30 (1935)
9. 9.
Hsu, T., Logan, M., Shahriari, S.: Methods for nesting rank 3 normalized matching rank-unimodal posets. Discrete Math. 309(3), 521–531 (2009)
10. 10.
Wang, Y.: Nested chain partitions of LYM posets. Discrete Math. 145(3), 493–497 (2005)