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 $(R,\mathfrak{m})$ and $\mathfrak{q}$ an ideal generated by a system of parameters $\underline{x} = (x_1,\ldots , x_d)$ of $M$. For each positive integer $n$, set

$$\Lambda_{d,n}=\{ \alpha =(\alpha_1,\ldots,\alpha_d)\in\mathbb{Z}^d\vert\alpha_i\geqslant 1, 1\leqslant i\leqslant d$$     and  $$\sum\limits_{i=1}^d\alpha_i=d+n-1\}$$

and $\mathfrak{q}(\alpha)=(x_1^{\alpha_1},\ldots,x_d^{\alpha_d})$ for each $\alpha\in\Lambda_{d,n}$. 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 $\underline{x}$ such that the equality $\mathfrak{q}^nM=\bigcap\limits_{\alpha\in\Lambda_{d,n}}\mathfrak{q}(\alpha)M$ holds true for all $n\ge1$. 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.