Multiset combinations (Mayor Percy Alexander MacMahon)

from math import factorial
from itertools import combinations
from functools import partial

def binomial_coefficient(elements, extraction):
    return factorial(elements)/(factorial(extraction)*factorial(elements-extraction))

def ℘(elements):
    powerset=[]
    for item in range(0,len(elements)+1):
        for subset in list(combinations(elements,item)):
            powerset.append(subset)
    return powerset

def negate(exponent):
    return (-1)**exponent

def combinatorial_subroutine(elements,extraction,outer_item,multiplicities):
    if elements+extraction-outer_item-sum(multiplicities)>0:
        return binomial_coefficient(elements-1+extraction-outer_item-sum(multiplicities),elements-1)
    else:
        return 0

def ΣΣ(inner_iterable,outer_iterable,inner_function,outer_function):
    inner_sum=outer_sum=0
    for outer_item in outer_iterable:
        for inner_item in inner_iterable:
            inner_sum+=inner_function(outer_item,inner_item)
        outer_sum+=outer_function(outer_item)*inner_sum
        inner_sum=0
    return outer_sum

def multiset_combinations(extraction,*multiplicities):
    elements=len(multiplicities)
    partial_combinatorial_subroutine=partial(combinatorial_subroutine,elements,extraction)
    return ΣΣ(℘(multiplicities),list(range(elements+1)),partial_combinatorial_subroutine,negate)

print(multiset_combinations(7,1,1,2,2,2,8))
input()