Refined enumeration of alternating sign matrices: Alternating sign matrices are generalizations of permutation matrices. Instead of allowing just 0's and 1's in prescribed arrangements, we now allow 0's, 1's and -1's with some additional restrictions. Determining the number of n-by-n alternating sign matrices was a big open problem solved independently by Zeilberger and just after by Kuperberg. Kuperberg's proof makes clever use of generating functions made from models for interacting particles, and we'll take that as a starting point for doing some more refined enumerations of generalizations of alternating sign matrices. There's a vast literature here (searching on the arXiv for "alternating sign matrices" yielded 169 results) but the following two papers of Kuperberg are a great place to start:

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• G. Kuperberg, Another proof of the alternating-sign matrix conjecture. Internat. Math. Res. Notices 1996, no. 3, 139–150. 

• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof. Ann. of Math. (2) 156 (2002), no. 3, 835–866.