solving it the convoluted way

2026-02-07

why do \(\binom{2n}{n},\) when I can just,

>>> import math
...
... def nPr(n, r):
...     return math.factorial(n) // math.factorial(n - r)
...
... def nCr(n, r):
...     return math.comb(n, r)
...
... def grid_formula(n):
...     N = (n + 1)**2 - 1
...     r = 2*n + 1
...     numerator = 2 * nPr(N, r)
...     denominator = nCr(N, r) * math.factorial(n) * math.factorial(n + 1)
...     return numerator // denominator
...
... print(grid_formula(19))
...
137846528820

The monster counts all possible permutations of grid vertices and then gets rid of permutations that are identical and normalizes the permutation explosion to fit my needs.

I wanted to solve this using permutations instead of combinations, ended up being more much complicated than I anticipated.