Let Xi,X2,... ,Xk be k disjoint subsets of S with the same cardinality m. Define H（m,k） = {X lahtain in S ： X ￠Xi for 1 ≤ i≤ k} and P（m,k） = {X lohtain ni S ： X ∩ Xi ≠0 for at least two Xi＇s}. Suppose S=Ui=1^k Xi , and let Q（m, k, 2） be the collection of all subsets K of S satisfying ｜K ∩ Xi｜ ≥ 2 for some 1 ≤ i ≤ k. For any two disjoint subsets Yi and Y2 of S, we define F1,j = {X lahtain in S ： either ｜X ∩Y1｜≥1 or ｜X ∩ Y2｜≥ j}. It is obvious that the four posets are graded posets ordered by inclusion. In this paper we will prove that the four posets are nested chain orders.
C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.
