How can I calculate simplex numbers in all dimensions?

dimension:     2|      3|      4|
simpl(1)       1|      1|      1|
simpl(2)       3|      4|      5|
simpl(3)       6|     10|     15|
simpl(4)      10|     20|     35|
simpl(5)      15|     35|     70|

First I considered about a recursive implementation (in python):

def simplex(n, dim):
    if dim == 1:
        return n
        i = 1
        ret = 0
        while i <= n:
            ret += simplex(i, dim-1)
        return ret

The algorithm works, but because of the recursion its really slow, especially if your dimension number is 5 or higher.


You should use a for loop rather than a while loop to solve the problem. This is as you're just re-writing a for loop. Using a for loop you can change the loop to a comprehension, and take the sum of it. This allows you to get the code:

def simplex(n, dim):
    if dim == 1:
        return n
    dim -= 1
    return sum(simplex(i, dim) for i in range(1, n + 1))

However this can be simplified. When dim is the bullet point the function is:

  1. \$n\$
  2. \$\Sigma_{i = 1}^{n} i\$
    \$= \frac{n(n + 1)}{2}\$
  3. \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} j)\$
    \$= \frac{n(n + 1)(n + 2)}{6}\$
  4. \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} (\Sigma_{k = 1}^{j} k)))\$
    \$= \frac{n(n + 1)(n + 2)(n + 3)}{24}\$
  5. \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} (\Sigma_{k = 1}^{j} (\Sigma_{l = 1}^{k} l))))\$
    \$= \frac{n(n + 1)(n + 2)(n + 3)(n + 4)}{120}\$

Since we're getting a pattern, we know that the top part of the fraction is:

\$\Pi_{i = 0}^{dim - 1} n + i\$

And if we use oeis to find the denominator it says it's probably the factorial sequence. And so it's likely:

\$\frac{\Pi_{i = 0}^{dim - 1} n + i}{dim!}\$

This is actually really easy to write in Python. And so you can get a significant speed up.

from functools import reduce
from math import factorial
from operator import mul

def simplex(n, dim):
    return reduce(mul, (n + i for i in range(dim))) // factorial(dim)
  • \$\begingroup\$ Good point, but I actually knew the iterative solution already at the time of asking the question (I just asked to share this in the community). As you can see, I answered the question too. I have two mentions to you: do you need the double Slash before factorial(dim)? and I like the while loop more than the for loop, because I expierienced weird things with the order of execution in for loops \$\endgroup\$ – Aemyl Sep 30 '16 at 5:41
  • \$\begingroup\$ @Aemyl I didn't see your answer when I started and finished mine, but it was a thought provoking question, :). To answer your 'mentions', (1) You need either // or /, I chose // as it matches the functions range. (2) I don't know how I can help you like for loops, but I will not retract my encouragement for you to use them. As they objectively make the function much cleaner than the equivalent using a while. \$\endgroup\$ – Peilonrayz Sep 30 '16 at 8:28
  • \$\begingroup\$ I didn't want you to retract your aspect of using for loops :) I just wanted to explain why I use while loops. \$\endgroup\$ – Aemyl Sep 30 '16 at 16:55
  • consistency: spaces around operators (esp. +=) is not always the same, you should stick to a space before and a space after.
  • conditions: since you return from your first if, you don't need the else clause. This saves an indentation level and help distinguish the general algorithm from the special case.
  • iteration: you don't need a while loop with an explicit index. for loops are prefered in python.

    for i in range(1, n + 1):
        ret += simplex(i, dim - 1)
  • additions: the sum builtin function should be faster than an explicit accumulation loop. You can combine it with a generator expression for better efficiency.

Overall, the code could be written:

def simplex(n, dim):
    if dim == 1:
        return n

    return sum(simplex(i, dim-1) for i in range(1, n+1))

As regard to the efficiency part, since your are computing results for the lower dimensions several times, you may want to use some form of caching. Take a look at the several techniques of memoization.

  • \$\begingroup\$ though for loops are prefered in python, I like while loops more then them, because in my opinion (and in python's) explicit is better than implicit. Another point is, I expierienced some really weird things with for loops (the order of iterating through list for example) \$\endgroup\$ – Aemyl Sep 30 '16 at 5:51
  • \$\begingroup\$ @Aemyl Out of "use <name> as a variable to iterate over <stuff> and do …" and "check this <condition> before you do … over and over again"; I don't think the latter is more explicit when, well, iterating over some stuff. And I don't know what you are trying to do with your lists, but order of iteration is always consistent in a for loop. I’d understand if you said set or dict, but not list. \$\endgroup\$ – 301_Moved_Permanently Sep 30 '16 at 7:58
  • \$\begingroup\$ sorry, I guess it was on dictionaries \$\endgroup\$ – Aemyl Sep 30 '16 at 16:53
  • \$\begingroup\$ @Aemyl Then the arbitrary ordering is built into the language. If you want to be able to control it, you may want to have a look at collections.OrderedDict. \$\endgroup\$ – 301_Moved_Permanently Oct 1 '16 at 6:24

Fortunatly, there is a iterative variant based on a sum formula by Gauss (the Gauss formula is only for the 2nd dimension):

def fac(n):
    i = 1
    e = 1
    while i <= n:
    return e

def simplex(n, dim):
    i = 0
    ret = 1
    while i < dim:
    ret /= fac(dim)
    return ret

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