  Let $(R,m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be
$m$-primary ideals in $R.$ In this paper we study the finiteness properties of
the sets $\Lambda_i^K(R):=\{g_i^K(Q): Q$ is a parameter ideal of $R\},$ where
$g_i^K(Q)$ denotes the Hilbert coefficients of $Q$ with respect to $K,$ for $1
\leq i \leq d.$ We prove that $\Lambda_i^K(R)$ is finite for all $1\leq i \leq
d$ if and only if $R$ is generalized Cohen-Macaulay. Moreover, we show that if
$R$ is unmixed then finiteness of the set $\Lambda_1^K(R)$ suffices to conclude
that $R$ is generalized Cohen-Macaulay. We obtain partial results for $R$ to be
Buchsbaum in terms of $|\Lambda_i^K(R)|=1.$ We also obtain a criterion for the
set $\Delta^K(R):=\{g_1^K(I): I$ is an m-primary ideal of $R\}$ to be finite,
generalizing preceding results.
