On the Conjecture of the Nested Chain Decomposition of the Normalized Matching Posets

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 的情況下是未解的。 在這篇論文裡，我們回顧了前人的結果與方法，提出一些新的技巧來證實更多 的秩為三的偏序集存在套鏈分解。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.List of Figures iv List of Tables iv 1. Introduction 1 2. Known results 5 3. Graded posets of width 14 and 15 8 4. Main theorems 11 5. Conclusion 18 References 1