The Garden Problem

Discrete Mathematics Level pending

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 \(n\).


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