Sum-of-Product of subsets

Question

Is there a name for this operation? And: is there a closed-form expression?

• For a given set of n elements, and value k between 1 and n,
• Take all subs

Answer 1

These are elementary symmetric polynomials. You can write them using summation signs as in Wikipedia. You can also use Vieta's formulas to get all of them at once as coefficients of a polynomial (up to signs)

(x-a_1)(x-a_2)...(x-a_k) =
   x^k -
   (a_1 + ... + a_k) x^(k-1) +
   (a_1 a_2 + a_1 a_3 + ... + a_(k-1) a_k)) x^(k-2) +
   ... +
   (-1)^k a_1 ... a_k

By expanding (x-a_1)(x-a_2)...(x-a_k) you get a polynomial time algorithm to compute all those numbers (your original implementation runs in exponential time).

Edit: Python implementation:

from itertools import izip, chain

l = [2,3,4]

x = [1]    
for i in l:
    x = [a + b*i for a,b in izip(chain([0],x), chain(x,[0]))]
print x

That gives you [24, 26, 9, 1], as 2*3*4=24, 2*3+2*4+3*4=26, 2+3+4=9. That last 1 is the empty product, which corresponds to k=0 in your implementation.

This should be O(N²). Using polynomial FFT you could do O(N log² N), but I am too lazy to code that.