Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules

Let M be a finitely generated module of dimension d over a Noetherian local ring ( R, m) and q an ideal generated by a system of parameters (x) under bar = (x(1),..., x(d)) of M. For each positive integer n, set Lambda(d,n) = {alpha = (alpha(1),....,alpha(d)) is an element of Z(d)vertical bar alpha(i) >= 1, 1 <= i <= d and Sigma(d)(i=1)alpha(i) = d + n- 1} and q(alpha) = ( x(1)(alpha 1),..., x(d)(alpha d)) for each alpha is an element of Lambda(d,n). Then we prove in this note that M is a sequentially Cohen- Macaulay module if and only if there exists a good system of parameters x such that the equality q(n)M = boolean AND(alpha is an element of Lambda d, n) q(alpha)M holds true for all n >= 1. As an application, we show that the sequentially Cohen-Macaulayness of a module can be characterized by a very special expression of the Hilbert-Samuel polynomial of a good parameter ideal.