Count stacking sequences

Question

Given a list of stack heights, calculate the number of ways those heights could have been arrived at by stacking blocks one at a time. Shortest code wins.

Test cases:

[2, 2] 6
[3,3,3] 1680
[1,5,9,1] 480480
[] 1
[10] 1
[2,2,2,10] 720720

Reference Python implementation:

def stacking_count(stacks):
     if sum(stacks) == 0:
         return 1
     count = 0
     for i, n in enumerate(stacks):
         if n == 0:
             continue
         stacks[i] -= 1
         count += stacking_count(stacks)
         stacks[i] += 1
     return count

Answer 1

Python 3, 85 bytes

f=lambda a:sum(f(a[:i]+[a[i]-1]+a[i+1:])for i in range(len(a))if a[i])if sum(a)else 1

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Answer 2

Jelly, 6 bytes

;@S!:/

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-7 bytes thanks to Jonathan Allan