Are there notations for sum over permutations?

Question

I am constructing combinatorial objects that have cyclic symmetry. I cannot remember if there is already a convention to indicate this. For example, if I have the sum

s = a₁b₂b₃ + a₂b₃b₁ + a₃b₁b₂,

I want to be able to indicate the summation over permutations of terms of order 3. I could try building something out of Σ and Π but it would be nice to have something neater. I can't remember if I learned this before.

Answer 1

We can write this expression using cyclic sum notation \begin{align*} \color{blue}{\sum_{\mathrm{cyc}}a_1b_2b_3}&=a_1b_2b_3+a_2b_3b_1+a_3b_1b_2\\ \end{align*} replacing the variables $a_1,b_2,b_3$ systematically in a cyclic manner.