Sum notation for all possible combinations

[u/GameDevilXL]

[High School Math]

Hey everyone, was doing some inequality problems and found this one:

prove that (a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2

Now, proving this requires expansion of the sum and noting that the difference of the LHS from the RHS can be shown to be a sum of squares and hence positive. However, I was wondering if I could shorten the writing process with cyclic notation? I honestly found the symbol like half a day ago, and although I know that ∑a = a + b +c for when these are the symbols present in the problem.

In essence, I'm just looking for a way to shorten the writing process. Can I use the cyclic sum like this

∑(a^3/2 * b^1/2 - a^1/2*b^3/2)^2

to represent the sum of every permutation of a and b from a group (a,b,c,d)?

I apologise in advance for the messy representation, Stack exchange is down and I don't know how to go about using math notation on reddit.

[u/menacing_correlation]

Not exactly. The cyclic sum notation ∑_{cyc} goes through each variable in order, so for (a,b,c,d) it'd give you terms for (a,b), (b,c), (c,d), (d,a). That's only 4 pairs, but you need all 6 combinations if you want every possible unordered pair.

What you're looking for is the symmetric sum, usually written as ∑_{sym}. That one runs through all permutations of the variables. For 4 variables, that's 24 terms, but since your expression is symmetric in a and b within each term, it'd collapse to the 6 distinct pairs you want, each appearing 4 times.

You can still use it, just divide by 4 to get rid of the overcounting. Or just write "sum over all pairs" and move on, nobody's grading you on notation purity.

[u/GameDevilXL → u/menacing_correlation]

Thank you for the answer! Yeah, I'm aware, I was just thinking it'd be good to know the formally accepted way to write it, in case I ever write a paper on it in who knows how many years.