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In this article, we prove that the Buchsbaum-Rim function$\ell_A(\S_{\nu+1}(F)/N^{\nu+1})$ of a parameter module $N$ in $F$ is boundedabove by $e(F/N) \binom{\nu+d+r-1}{d+r-1}$ for every integer $\nu \geq 0$.Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once theequality holds for some integer $\nu$. As a direct consequence, we observe thatthe first Buchsbaum-Rim coefficient $e_1(F/N)$ of a parameter module $N$ isalways non-positive.

A Note on the Buchsbaum-Rim function of a parameter module