MathDB

Each cell of an

m\times n

board is filled with some nonnegative 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

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

Problem Statement