Given a set of positive integers, and a non-negative target sum, find all the combinations of the numbers from the set that sum up to the target sum, order is significant. For example, for {1,2}, 5, there is 8 combinations.
Using polynomes to represent different sums and lengths of combinations. Each polynome represents the numbers of ways to sum up to all possible sums, depending on the length of the combination. For each length, there is one polynome that will contain all the counts for each sum that's accessible through a combination. For example, the polynome P(x)=(x^2+x)(x^2+x) represents a combination of two numbers from the set {1,2}. When expanded, we get P(x)=x^4+2*x^3+x^2. In this form, the coefficients are the number of combinations, and the matching monome power is the sum for which the number of combination is counted. From the above example, there's 1 way to sum to 4, 2 ways to sum to 3 and 1 way to sum to 2, using a combination of 2 numbers from the set {1,2}.
In [1]: poly_sum_ordered_reps([1,2,3],1000)
Out[1]:
2758842807766486252615892411656158645133100149652696210351601845036392978912293462801016485671033253921841350537004356434253826361707295202024537559785200706502368152965047761644352316799391470273906561574500883480570560512982435681502330814068718832813973880527601
In [2]: