  Let $A$ be a subset of positive integers. By $A$-partition of $n$ we
understand the representation of $n$ as a sum of elements from the set $A$. For
given $i, n\in\N$, by $c_{A}(i,n)$ we denote the number of $A$-partitions of
$n$ with exactly $i$ parts. In the paper we obtain several result concerning
sign behaviour of the sequence
$S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{i}i^{k}c_{A}(i,n)$, where $k\in\N$ is fixed. In
particular, we prove that for a broad class $\cal{A}$ of subsets of $\N_{+}$ we
have that for each $A\in \cal{A}$ we have $(-1)^{n}S_{A,k}(n)\geq 0$ for each
$n, k\in\N$.
