Coefficient values in functions,

[spatzbw]

I noticed these patterns when calculating some co-efficents in functions having this pattern. I would appreciate a better understanding of it, so if you can put in any input that may help, that would be awesome. Also, I tried to explain the pattern as best as I can.

Let a set A = {a1, a2, ..., an}, where a1 < a2 < a3 < ... < an. Let A:k:n be as such; To present the idea, let n =4. so A:1:4 = a1 + a2 + a3 + a4, A:2:4 = (a1)(a2) + (a1)(a3) + (a1)(a4) + (a2)(a3) + (a2)(a4) + (a3)(a4), A:3:4 = (a1)(a2)(a3) + (a1)(a2)(a4) + (a1)(a3)(a4) + (a2)(a3)(a4) and A:4:4 = (a1)(a2)(a3)(a4). So if A = {1,2,3,4,5}, A:1:5 = 15, A:2:5 = 2+3+4+5+6+8+10+12+15+20, and so on. Keep in mind that since for example in A:3:n, (ai)(am)(an), where i,m,n are all natural numbers less than or equal to n, uses the same terms from A, we do not reuse them as (am)(ai)(ak) or any other variation of those 3 numbers. My question is, is there a symbol that refers to this, if so what is it, and what math area does this most directly fall under so I an further research it. Thank you

 

[micromass]

Hi spatzbw!

The following article will interest you: http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial

Basically, the things you calculate are simply values of elementary symmetric polynomials...