# Combinatory sum of multiplications

Suppose i have $N$ variable. In a sum, i have terms each consist of combination of n variable. Each variable(they appear only once in one term) is to be multiplied to get term. How can i write the sum in a compact way (in terms of sigma maybe)? Example to be clear:

Let $S_{N,n} = S(a_1,a_2,a_3...,a_N)$ be our sum:

$S_{3,1} = S(a_1,a_2,a_3) = a_1 + a_2 + a_3$
$S_{3,2} = S(a_1,a_2,a_3) = a_1a_2 + a_1a_3 + a_2a_3$
$S_{3,3} = S(a_1,a_2,a_3) = a_1a_2a_3$

$S_{4,2} = S(a_1,a_2,a_3,a_4) = a_1a_2 +a_1a_3 +a_1a_4 +a_2a_3 +a_2a_4 +a_3a_4$

• Not sure what you are asking. You give three conflicting definitions of $S(a_1,a_2,a_3)$. Your definition of $S_{N,n}$ contains neither an $N$ nor an $n$. Are you asking for a characterization of the symmetric polynomials in $n$ letters? If so, see this
– lulu
Apr 11, 2016 at 20:11
• @lulu $n$ means terms are going to be group of $n$ variable multiplied, where there exist $N$ variable. Apr 11, 2016 at 20:15
• Ok, but then your edit is wrong (as you only have $n$ variables). And the "definition" you give doesn't define anything. But, I think I understand that you are looking at the elementary symmetric polynomials in $N$ letters. Up to sign they are the coefficients of $\prod_{i=1}^N (x - a_i)$. The link I gave in my first comment might be helpful.
– lulu
Apr 11, 2016 at 20:18
• If you are just looking for a formal way to write the expression, how about $$S_{N,n}=\sum_{1≤i_1<i_2\cdots<i_n≤N}\;a_{i_1}a_{i_2}\cdots a_{i_n}$$
– lulu
Apr 11, 2016 at 20:23
• Well...sure. You could split off the terms that have an $a_1$ in them, so $$S_{N,n}(a_1,a_2,\cdots, a_N)=a_1\times S_{N-1,n-1}(a_2,\cdots, a_N)+S_{N-1,n}(a_2,\cdots,a_N)$$ That's not a bad recursion. Helpful if you are programming a lot of these. Not sure it's very intuitive though.
– lulu
Apr 11, 2016 at 20:30

Let $N=\{ 1,...,m\}$ (the first $m$ naturals) be the number of variables, and $n$ be how many distinct variables appear in each term. Let $T\subset N$, then
$S_{N,n}=\underset{|T|=n}{\sum}\underset{t\in T}{\prod}a_t$