Strange summation I don't understand the following notation
\begin{equation}F(\xi_1, \dots, \xi_k) \triangleq \sum_{1\leq i_1 \neq \dots \neq i_k \leq m}f_{i_1}(\xi_1)\cdots f_{i_k}(\xi_k)\end{equation}
such summations ''are taken over all distinct $i_1,\dots,i_k$ such that $1 \leq i_1, \dots, i_k \leq m$''. I know that all index $i_1,\dots,i_k$ are integers and I know that $k$ is an integer that ranges between $1$ and $m$. The integer $m$ is a given parameter.
This kind of summations arise in the theory of multi-Bernoulli random finite sets , for example see equation (11) of this paper.
 A: In the paper it says it's summing over the permutations. You have M different functions to play with. K must range between 1 and M because there are only that many distinct integers between 1 and M. Thus some combinations only have 1 index to worry about, others have 2, 3, ... M.
You're allowed to take any distinct set of integers between 1 and M. Then multiply the functions corresponding to those groups of distinct integers. So if you have k = 3, and $i_{1} = 1, i_{2} = 3, i_{3} = 5$, then inside the summation for that term is $f_{i_{1}}*f_{i_{2}}*f_{i_{3}} = f_{1}*f_{3}*f_{5}$
However im pretty sure that you could still have k = 3 and have $i_{1} = 3, i_{2} = 5, i_{3} = 7$ or any value for each index as long as the values are bounded by M.
The idea is that the summation is trying to include every value of k, and all the possible permutations for each value of k.
The interesting part of this notation is that all the functions are in terms of different input parameters. $\xi_{1}$ through $\xi_{K}$. So that the function you label $f_{k}$ always takes the input $\xi_{k}$.
To really drill it home:
Sometimes the use of the function will look like F($\xi_{1}$), which means k =1.  And you just add up all the functions with the value $\xi_{1}$ as input.
So
$F(\xi_{1}) = f_{1}(\xi_{1}) + f_{2}(\xi_{1}) + ... + f_{M}(\xi_{1}). $
or maybe it will look like $F(\xi_{1}, \xi_{2}, ... \xi_{k})$ which then includes all the permutations of the functions 1 through M too but each term only include k functions.
Tl;DR: All its saying is that you add up the product of every possible subcollection of functions between $f_{1}$ and $f_{M}$
