Summing pairs from a sequence

Question

Suppose we have an arbitrary sequence $$\{a_k\} = \{a_1, a_2, ..., a_k\} $$

and use it to a create a set as follows

$$A = (a_i+ a_j : a_i, a_j \in \{a_k\})$$

and we wanted to sum over all of the members of this set, would we denote it as $\sum A$ ?

Furthermore, if we wanted to create a new set $A^*$ which excludes terms $a_i + a_j$ if $i=j$, how could we write that more succinctly?

How about if we wanted to exclude terms $a_j + a_i$ aswell? i.e. $a_i + a_j$ in reverse order?

Ultimately, I'm looking for notation akin to summing over all "unique pairwise sums" of the members of a set (something like $\sum B$ where $b := a_i + a_j$ and $b \in B$) excluding the cases where $i=j$, and excluding all the "duplicate pairwise sums" of the form $a_j + a_i$ given that they're not "unique" ($a_j + a_i$ is given to be "equivalent" to $a_i + a_j$)

I'm assuming there is a more succinct way to write "the sum of all unique pair-wise sums of the members of a set", because I vaguely remember notation form probability theory to denote "all pairs from a set where order doesn't matter"... something to do with binomials if I remember correctly.

Apologies in advance for my drawn out description... Any better notation that captures what I'm trying to capture would be much appreciated.

Answer 1

and we wanted to sum over all of the members of this set, would we denote it as $\sum A$?

Yes this is fine notation. You could also write $\sum_{a \in A} a$ or $\sum_{i = 1}^k a_i$.

Furthermore, if we wanted to create a new set $A^*$ which excludes terms $a_i + a_j$ if $i = j$, how could we write that more succinctly?

$A^* = \{a_i + a_j : a_i, a_j \in \{a_k\} \text{ and } i \neq j\}$.

How about if we wanted to exclude terms $a_j + a_i$ as well? i.e. $a_i+a_j$ in reverse order?

I'd denote this $\{a_i + a_j : a_i, a_j \in \{a_k\} \text{ and } i < j\}$. This will avoid this double counting of sums as well as the $a_i + a_i$.

I'm assuming there is a more succinct way to write "the sum of all unique pair-wise sums of the members of a set"

In the case here, I'd write $\sum_{i < j} a_i + a_j$. The same works for any set indexed by something totally ordered.