Let $S$ be a set of numbers of size $n$.
Let $M_S$ be the set of all multisets of size $n$ made of elements from $S$, with at least one element repeated. For example, if $S=\{1, 2, 3\}$, then $M_S = \{$ $\{1, 1, 1\}$ $\{2, 2, 2\}$ $\{3, 3, 3\}$ $\{1, 1, 2\}$ $\{1, 1, 3\}$ $\{1, 2, 2\}$ $\{2, 2, 3\}$ $\{1, 3, 3\}$ $\{2, 3, 3\}$ $\}$
The $\bf{condition}$ that $S$ should satisfy is that there should be no multiset in $M_S$ the sum of whose elements is equal to the sum of elements of $S$. For Example: $S_1 = \{1, 2, 3, 4, 5, 6\}$ does not satisfy the condition as $sum(\{1, 2, 1, 6, 5, 6\})$ = $sum(\{1, 2, 3, 4, 5, 6\})$ while $S_2 = \{1, 7, 43, 259, 1037, 6223\}$ does satisfy the condition.
Now, let $U$ be the set of all sets(such as $S_2$) satisfying the above $\bf{condition}$ for a given $n$. We need any one set $T$ from $U$ such that sum of elements of $T$ is minimum across all sets in $U$.
Q1. How can we generate $T$ ?
Q2. How does the sum of elements in $T$ grow as a function of $n$ asymptotically ?
