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Problem 106

Special subset sums: meta-testing

Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:

1. $S(B) \neq S(C)$; that is, sums of subsets cannot be equal.
2. If $B$ contains more elements than $C$ then $S(B) > S(C)$.

For this problem we shall assume that a given set contains n strictly increasing elements and it already satisfies the second rule.

Surprisingly, out of the $25$ possible subset pairs that can be obtained from a set for which $n = 4$, only $1$ of these pairs need to be tested for equality (first rule). Similarly, when $n = 7$, only $70$ out of the $966$ subset pairs need to be tested.

For $n = 12$, how many of the $261625$ subset pairs that can be obtained need to be tested for equality?

NOTE: This problem is related to Problem 103 and Problem 105.

特殊的子集和：元检验

1. $S(B) \neq S(C)$；也就是说，任意子集的和不相同。
2. 如果$B$中的元素比$C$多，则$S(B) > S(C)$。