Notation of symmetric sum notation When you use the symmetric sum notation, for example, $$\sum_\text{sym}abc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?
I am confused about repetitions of the same expression in a symmetric sum notation.

The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $\sum\limits_{\rm sym}$, but preferably written a $\sum\limits_\sigma$
$$\sum_{\sigma} f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$
So in this case $$\sum_{\sigma} abc = 3 abc$$ 

The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.
The first elemental symmetric sum is confussingly also indicated with the $\sum\limits_{\rm sym}$ operator. 
The $k$-th elemental symmetric sum is usually indicated with $\sum\limits_{\rm sym}^k$.   
And so, for example, $$\sum_{\rm sym}^2 \{a,b,c\} = ab + ac+ bc$$

