Notation of symmetric sum notation

Question

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.

Answer 1

I think if we want to work with symmetric sum, it's better to write $$\sum_{sym}(abc+a)=6abc+2(a+b+c)$$ and $$\sum_{sym}abc+a=6abc+a.$$

Answer 2

We have that

$$\sum_\mathrm{sym}Q(x_i)=\sum_\sigma Q(x_{\sigma(i)})$$

for all permutations of $1, \ldots , n$.

Therefore it should be

$$\sum_\text{sym}abc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$

$$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$

Answer 3

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 $$\sum_{\sigma} abc = 3 abc$$

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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, $$\sum_{\rm sym}^2 \{a,b,c\} = ab + ac+ bc$$

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Answer 4

The symmetric sum notation $\sum_{\color{blue}{\mathrm{sym}}}$ is the sum over all permutations of the elements of a predefined set $S$.

• If $S=\{a,b,c\}$, then \begin{align*} \color{blue}{\sum_{\mathrm{sym}}\left(abc+a\right)}&=(abc+a)+(acb+a)+(bac+b)\\ &\quad+(bca+b)+(cab+c)+(cba+c)\\ &\,\,\color{blue}{=6abc+2(a+b+c)} \end{align*}

• If $S=\{a,b\}$, then \begin{align*} \color{blue}{\sum_{\mathrm{sym}}\left(abc+a\right)}&=(abc+a)+(bac+b)\\ &\,\,\color{blue}{=2abc+a+b} \end{align*}

On the other hand the cyclic sum notation $\sum_{\color{blue}{\mathrm{cyc}}}$ is the sum over elements of a predefined set $S$ in a cyclic manner $a\to b\to c\to\cdots\to a$.

• If $S=\{a,b,c\}$, then \begin{align*} \color{blue}{\sum_{\mathrm{cyc}}\left(abc+a\right)}&=(abc+a)+(bca+b)+(cab+c)\\ &\,\,\color{blue}{=3abc+a+b+c} \end{align*}
• If $S=\{a,b\}$, then \begin{align*} \color{blue}{\sum_{\mathrm{cyc}}\left(abc+a\right)}&=(abc+a)+(bac+b)\\ &\,\,\color{blue}{=2abc+a+b} \end{align*}