摘要
对称键是一种特殊的偏序，用它已经得到了许多非常漂亮的结果．如果一个偏序集可以分解成不相交的对称链之并，则称此偏序集具有对称链分解．但目前已证明具有这种分解的偏序集并不多．Ｌ（ｍ，ｎ＝｛（ｘ_１，ｘ_２，…，ｘ_ｎ）ｘ_ｉ均为整数且０≤ｘ_１≤ｘ_２≤…ｘ_ｍ≤ｎ｝，序关系≤定义为：Ｘ＝（ｘ_１，ｘ_２，…，ｘ_ｍ）≤Ｙ＝（ｙ_１，ｙ_２，…，ｙ_ｍ）充要条件是对所有ｉ，ｘ_ｉ≤ｙ_ｉ．有人猜测Ｌ（ｍ，ｎ）具有对称链分解．１９８０年，Ｌｉｎｄｓｔｒｏｍ和Ｗｅｓｔ分别证明了Ｌ（３，ｎ），Ｌ（４，ｎ）猜想成立．本文构造性地证明了对于Ｌ（ｍ，１），Ｌ（ｍ，２），Ｌ（ｍ，３）猜想成立，并讨论了有关计数问题．
Symmetric chain is a special partial order. Many beautiful results with it have been obtained a poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. But there have not been so many such kind of posets so far. L(m,n) = { (x_1,x_2, …,x_m)xi integers, 0 ≤x_1 ≤x_2 ≤ … ≤x_m ≤ n } with order relation ≤ defined by X = (x_1,x_2, …,x_m) ≤Y = (y_1,y_2, …,y_m) iff x_i ≤y_i for each i. It has been conjectured that each L(m,n) is a symmetric chain decomposition. At present, the conjecture has been confirmed only for L(3,n) by Lindstrom (1980) and for L(4,n) by West (1980). This paper proves that the conjecture is true for L(m, 1),L(m,2) and L(m, 3), and corresponding counting is discussed.
