L(m,3)的对称链分解

Symmetric Chain Decomposition of L(m,3)

中文关键词:  序  对称链  对称链分解.

英文关键词:order  symmetric chain  symmetric chain decomposition.

中文摘要:

      对称链是一种特殊的偏序,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)＝{(x₁,x₂,…,x_n)|x_i均为整数且0≤x₁≤x₂≤…≤x_m≤n},序关系≤定义为:X=(x₁,x₂,…,x_m<

英文摘要:

      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) = { (x₁,x₂,…,x_m)x_i integers, 0 ≤x₁ ≤x₂≤ … ≤x_m≤ n } with order relation ≤ defined by X = (x₁,x₂,…,x_m) ≤Y = (y₁,y₂,…,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),L(m,3), and corresponding counting is discussed.