B6

Part of 2023 Putnam

Problems(1)

Source:

n

be a positive integer. For

i

j

\{1,2, \ldots, n\}

s(i, j)

be the number of pairs

(a, b)

of nonnegative integers satisfying

a i+b j=n

S

n

-by-n matrix whose

(i, j)

s(i, j)

. For example, when

n=5

S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]

. Compute the determinant of

S

PutnamPutnam 2023