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Omni-MATH / Each cell of an m×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 …

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problem

Each cell of an m×nm\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 00 or 11 . (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to 00 . Determine the number of distinct gardens in terms of mm and nn .
Plain-text mathematical notation (without MathML)
Each cell of an m×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 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 .
Original LaTeX notation
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 $0$ 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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Source and history

task_id
c0439dfe-0a35-54f9-884c-86e6eecc9d5b
task_key
test--c0439dfe-0a35-54f9-884c-86e6eecc9d5b
task_revision_id
1
config / split
/ test

Official source

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