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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:

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

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

Salihcyilmaz
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  • 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 '16 at 20:11
  • @lulu $n$ means terms are going to be group of $n$ variable multiplied, where there exist $N$ variable. – Salihcyilmaz Apr 11 '16 at 20:15
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    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 '16 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 '16 at 20:23
  • @lulu Im looking to your link you gave now. It seems they already developed an expression. Can we improve index notation they offer (i.e$ i<j<...< n$) which is kind of confusing. Im looking for something like concatenated Sigmas. – Salihcyilmaz Apr 11 '16 at 20:25
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    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 '16 at 20:30

1 Answers1

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I suspect you are wanting to know if there is a compact way to write Symmetric Polynomials, and afaik, the best you could w/ sigma notation would something like:

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

$S_{N,n}=\underset{|T|=n}{\sum}\underset{t\in T}{\prod}a_t$

Justin Benfield
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