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I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.

Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with $m,n$ positive integers. Define $N(m,n)$ to be the number of subdivisions of $L_{m,n}$ into lattice triangles of area $1/2$. Here a lattice triangle is a triangle with vertices on the lattice $\mathbb Z^2\subset\mathbb R^2$. Is there an explicit formula for $N(m,n)$?

For example, $N(1,1)=2$, $N(1,2)=6$...

Thank you very much in advance.

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  • $\begingroup$ It's not hard to see that $N(1,n) = {2n \choose n}$, but I guess you already knew that... $\endgroup$
    – Tom De Medts
    Commented Oct 2, 2014 at 15:24

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Have you seen this paper by Kaibel and Ziegler ? Their notion of unimodular triangulation appears to be the same as what you ask for.

They do not present explicit formulas, but rather discuss upper and lower bounds which are exponential in $mn$.

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