Count integer partitions with k parts when partition elements given

Question

I want to count the integer partitions of n with k partition elements. The possible partition elements are defined through a given vector v with distinct elements. Partition elements can be choosen more than once. How Can i do this? Optimal without iterating through all integer partitions of n.

Example:

n := 10

k := 3

v := 1,2,6,7,8

=> 3

Answer 1

One way is to have the recurrence consider each element in order.

Unmemoised JavaScript:

function f(n, k, v, i=0){
  if (k == 0)
    return n == 0;

  if (i == v.length)
    return false;
  
  let total = 0
  
  while (k >= 0 && n >= 0){
    total = total + f(n, k, v, i+1);
    k = k - 1;
    n = n - v[i];
  }
  
  return total;
}

console.log(f(10, 3, [1,2,6,7,8]));