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標題: 關於正則匹配偏序集之套鍊猜想的研究
On the Conjecture of the Nested Chain Decomposition of the Normalized Matching Posets
作者: 張育綸
Yu-Lun Chang
關鍵字: 偏序集;套鏈分解;正則匹配;Poset;nested chain decomposition;normalized matching
引用: [1] I. Anderson. Some problems in combinatorial number theory. Ph.D. Thesis. University of Nottingham.(1967). [2] I. Anderson. Combinatorics of Finite Sets. Dover Publications(2002). [3] R.P. Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics Second Series. 51, 161-166(1950). [4] E. G. Escamilla, A. C. Nicolae, P. R. Salerno, S. Shahriari, J. O. Tirrell. On Nested Chain Decompositions of Normalized Matching Posets of Rank 3. Order 28, 357-373(2011). [5] J.R. Griggs. Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math. 32, 807-809(1977). [6] T. Hsu, M. Logan, S. Shahriari. Methods for nesting rank 3 normalized matching rank-unimodal posets. Discrete Math. 309(3), 521-531(2009). [7] Y. Wang. Nested chain partitions of LYM posets, Discrete Math. 145(3), 493-497(2005).
令P 是正則匹配單峰偏序集。我們可將P分割成一堆鏈C1, C2, ..., Cn,並且稱這些鏈所形成的集合C 是P 的一個鏈分解。 若在C 裡的任兩條鏈Ci,Cj 滿足當Ci的長度小 於等於Cj 的長度會有Ci的元素的秩的集合是Cj 的元素的秩的集合的子集合的話, 就稱C是套鏈分解,若P 有這樣的鏈分解,就稱P 可被套鏈分解。
1975年 ,Griggs做了以下猜想,任何正則匹配單峰的偏序集都可被套鏈分解”。至今,只有在秩為二以及一些秩為三的偏序集被證實滿足該猜想,但在一般

Let P be a normalized matching rank-unimodal poset. We can partition P into
chains C1;C2;...;Cn, and name C = fC1;C2;...;Cng a chain decomposition of P.
The decomposition C is said to be nested, if any two different chains Ci,Cj in C
with length of Ci less than or equal to the length of Cj will imply the set of ranks
of elements in Ci is a subset of the set of those in Cj . If there exists such a chain decomposition in P, then P is nested.
In 1975, Griggs conjectured thatEvery normalized matching rank-unimodal poset is nested.' Till now, the conjecture is proved to be true for all posets of rank 2 and some posets of rank 3, but it is still widely open in general.
In this thesis, we will present some progress on the posets of rank 3.
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