Question 177
Let $m$ and $n$ be positive integers, and let $\mathcal{R}$ be a $2m \times 2n$ grid of unit squares.
A *domino* is a $1 \times 2$ or $2 \times 1$ rectangle. A subset $S$ of grid squares in $\mathcal{R}$ is *domino-tileable* if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside $S$. *Note*: The empty set is domino-tileable.
An *up-right path* is a path from the lower-left corner of $\mathcal{R}$ to the upper-right corner of $\mathcal{R}$ formed by exactly $2m + 2n$ edges of the grid squares.
Determine, with proof, in terms of $m$ and $n$, the number of up-right paths that divide $\mathcal{R}$ into two domino-tileable subsets.
A *domino* is a $1 \times 2$ or $2 \times 1$ rectangle. A subset $S$ of grid squares in $\mathcal{R}$ is *domino-tileable* if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside $S$. *Note*: The empty set is domino-tileable.
An *up-right path* is a path from the lower-left corner of $\mathcal{R}$ to the upper-right corner of $\mathcal{R}$ formed by exactly $2m + 2n$ edges of the grid squares.
Determine, with proof, in terms of $m$ and $n$, the number of up-right paths that divide $\mathcal{R}$ into two domino-tileable subsets.