Number partition subjected to the cardinality of subset

I hope someone can take some time to consider the following problem and welcome to discuss together.

Number partition problem is one of well-known NP-hard problems.

Now I am considering the hardness of constrained partition problem. In particular, the cardinality of multiset $$S$$ is n, denoted by $$|S|=n$$. Now is it NP-completeness to determine whether a given multiset $$S$$ of positive integers can be partitioned into two subsets $$S_1$$ and $$S_2$$ such that the sum of the numbers in $$S_1$$ equals the sum of the numbers in S2, and subjected to the cardinality of sub multiset $$S_1$$ is equals $$k$$, that is $$|S_1|=k$$ where $$k <= n$$

• Did you try the reduction from Number partition problem? Aug 29 '21 at 4:25
• Intuitively, i think the constrained number partition is harder than that original one Aug 29 '21 at 7:27
• You are right! For a formal proof, try to show a polynomial-time reduction from the Number partition problem to the constrained number partition problem. It is not that difficult. Aug 29 '21 at 7:46