Is there an alternate way to represent the multiple summation given below?
$\displaystyle \Large \sum_{i_k=k}^{n} \space \sum_{i_{k-1}=k-1}^{i_k} \dots \sum_{i_2=2}^{i_3} \space \sum_{i_1=1}^{i_2}$
It guess it is wrong to write it as a product of summations like $\displaystyle \large \prod_{\Large j=k}^{1}\space \sum_{\Large i_j=j}^{\Large i_{j+1}}$.
What's a better, concise notation ?
Update: Another way is $\displaystyle \quad \large \sum\limits_{\Large n \geq i_k \geq i_{k-1} \dots \geq i_2 \geq i_1 \geq 1}$.
Are there any other ways?
Best Answer
I would use $$ \large \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_k \le n \\ i_j \ge j \;\; \forall j}} $$
A summation in this form is understood to be taken over all ordered tuples $(i_1,i_2,\cdots,i_k)$ which satisfy the inequalities given. See Wikipedia for a list of various notational conventions.