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If we define our movements through a tiling domino tiling to be. Then through any tiling we can have these paths from our sources to our sinks.
These movements are similar to Schröder paths. For example, consider an Aztec Diamond of order 2, and after drawing its directed graph we can label its sources and sinks.
The number of tilings for order 2 is. According to Lindstrom-Gessel-Viennot , if we let S be the set of all our sources and T be the set of all our sinks of our directed graph then.
Considering the directed graph of the Aztec Diamond, it has also been shown by Eu and Fu that Schröder paths and the tilings of the Aztec diamond are in bijection.
Another way to determine the amount of tilings of the Aztec Diamond is using Hankel matrices of large and small Schröder numbers ,  using the method from Lindstrom-Gessel-Viennot again.
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