The Garden Problem
Each cell of an \(m\) by \(n\) board is filled with some non-negative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. Note that two numbers in cells that share only a corner are not adjacent. The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either 0 or 1.
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to 0.
Determine the number of distinct gardens in terms of \(m\) and \(n\).