Special Subset Sums: Optimum

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:
$S(B) \ne S(C)$; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S(B) \gt S(C)$.
If $S(A)$ is minimised for a given $n$, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.

$n = 1$: $\{1\}$
$n = 2$: $\{1, 2\}$
$n = 3$: $\{2, 3, 4\}$
$n = 4$: $\{3, 5, 6, 7\}$
$n = 5$: $\{6, 9, 11, 12, 13\}$
It seems that for a given optimum set, $A = \{a_1, a_2, \dots, a_n\}$, the next optimum set is of the form $B = \{b, a_1 + b, a_2 + b, \dots, a_n + b\}$, where $b$ is the “middle” element on the previous row.
By applying this “rule” we would expect the optimum set for $n = 6$ to be $A = \{11, 17, 20, 22, 23, 24\}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = \{11, 18, 19, 20, 22, 25\}$, with $S(A) = 115$ and corresponding set string: 111819202225.
Given that $A$ is an optimum special sum set for $n = 7$, find its set string.
NOTE: This problem is related to Problem 105 and Problem 106.

We will use a brute force approach to solve this problem, since the enumeration of sets is manageable if we follow the properties. For each size ‘n’, we will start with the “middle” element one step above the largest element from size ‘n-1’ set.

Let’s start by deriving the special sum sets for `n = 6` from `n = 5` = `{6, 9, 11, 12, 13}`:

We find the middle element from the above set and add 1 which is `12 + 1` = `13`. We then create a candidate set `{13, 19, 22, 24, 25, 26}`. We verify that this candidate set obeys the two properties which it does. Then, we try to minimize the set by subtracting 1 from the largest element and check if the set still respects the properties. The set `{13, 19, 22, 24, 25, 25}` does not respect the property, as two sums are equal, hence our candidate set is indeed the optimum special sum set for `n = 6`. The sum `S(A) = 13 + 19 + 22 + 24 + 25 + 26 = 129`.

Now, let’s find the optimum special sum set for `n = 7` following the same strategy:

We use the largest element from the `n = 6` set, that is `26`. We add `1` to `26` that gives `27`. We form a candidate set `{27, 40, 46, 49, 51, 52, 53}`. Following the same procedure to minimize the set, we see that `{27, 40, 46, 49, 51, 52, 52}` is not a valid set as it breaks the property. Therefore, our candidate set is indeed the optimum special sum set.

The set string for `n = 7` is then: 27404649515253.

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