Partition of a set of integers into subsets with prescribed sums I saw this problem: 

A non increasing sequence of positive integers $m_1,m_2,..., m_k$ is said to be n-realizable if the set $I_n=\{1,2,..., n\}$ can be partitioned into $k$ mutually disjoint subsets $S_1,S_2,...,S_k$ such that $\sum_{x\in S_i} x = m_i$ for each $1\leq i \leq k$. 

in the paper "PARTITION OF A SET OF INTEGERS INTO SUBSETS WITH
PRESCRIBED SUMS", by Fu-Long Chen, Hung-Lin Fu, Yiju Wang and Jianqin Zhou 

http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/1028

They have solved the problem under certain constraints. But I can't find anything about its complexity in general. Does anyone know a reference about the complexity of this problem?
It reminds me of the bin-packing problem, or in some sense, it is similar to the subset sum problem. So, I guess it must be NP-complete in general?

More precisely, I like to prove the NP-completeness for the fixed value of $k$, for example, when $k = 3, 4, \ldots$?
In this case, it is very similar to bin-packing or knapsack problem, but as we want the equality it is different. Maybe there are variations of these problems that match my question?
For any fixed $k$, isn't it in P, by dynamic programming?

For each $i\in\{0,\ldots,n\}$ and $t_1, t_2, ..., t_k$ such that each $t_i \in \{0,\ldots,m_i\}$, define $S(i, t_1, t_2, \ldots, t_k)$ to be true iff $(t_1, t_2, .., t_k)$ is $i$-realizable.  Then there are $O(n^{2k+1})$ such subproblems (assuming WLOG that $\max_i m_i\le n^2$), and you have a recurrence like

$ S(i, t_1, t_2, \ldots, t_k)$ 

$~ = S(i-1, t_1 - i, t_2, \ldots, t_k) $

$~\vee~ S(i-1, t_1, t_2 - i, t_3, \ldots, t_k)$

$~\vee~ \cdots$

$~\vee~ S(i-1, t_1, t_2 , t_3, \ldots, t_{k-1}, t_k- i)?$