In How is Euler Calculus Going to Change Compressive Sensing ? , Dan mentioned in the comment section that he had written something on this subject (in Haskell): Target Enumeration with the Euler Characteristic. Parts 1 & 2. With regards to the historical addendum of the post, looks to me like Andrew Ng is proving empirically that deep neural networks are on the contrary helping in the resurgence of AI (and dictionary learning for that matter). Of related interest, I note from this entry this connection with counting by John Baez.
Since I first wrote this entry, I have added additional references, here they are:
The third paper mentions another implementation in Java that will be released after review.