Symmetric Chain Decomposition of L(m,3)

Abstract:

      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.