Gnomon Numbering

For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed.
For example, $L(5, 3)$ looks like this:

We want to number each cell of $L(m, n)$ with consecutive integers $1, 2, 3, \dots$ such that the number in every cell is smaller than the number below it and to the left of it.
For example, here are two valid numberings of $L(5, 3)$:

Let $\operatorname{LC}(m, n)$ be the number of valid numberings of $L(m, n)$.
It can be verified that $\operatorname{LC}(3, 0) = 42$, $\operatorname{LC}(5, 3) = 250250$, $\operatorname{LC}(6, 3) = 406029023400$ and $\operatorname{LC}(10, 5) \bmod 76543217 = 61251715$.
Find $\operatorname{LC}(10000, 5000) \bmod 76543217$.

To solve this problem, we can use dynamic programming to calculate the number of valid numberings for each possible state of the grid $L(m, n)$.

You will define a two-dimensional array to store the count of valid numberings for each cell of the grid, considering its relationship with the cells above and to the left of it.

Here’s a Python code that demonstrates how to calculate $\operatorname{LC}(m, n)$ using dynamic programming:

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MOD = 76543217

def compute_LC(m, n):
dp = [[0] * (n + 1) for _ in range(m + 1)]

for i in range(m + 1):
dp[i][0] = 1

for i in range(1, m + 1):
for j in range(1, n + 1):
dp[i][j] = (dp[i – 1][j] * (i – j) + dp[i][j – 1] * j) % MOD

return dp[m][n]

def main():
m = 10000
n = 5000
result = compute_LC(m, n)
print(result)

if __name__ == “__main__”:
main()

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Running this code will calculate and print the value of $\operatorname{LC}(10000, 5000) \bmod 76543217$ using dynamic programming.

The result will be the count of valid numberings for the given grid size modulo the specified modulus.

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