You might wish to look at some papers by Paul Zinn-Justin. For example "Littlewood--Richardson coefficients and integrable tilings" defines a model of random tilings which count the Littlewood-Richardson coefficients. As early as 2001, puzzles were being used to compute the cohomology of complex Grassmanians. See "The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone" by Allen Knutson and Terry Tao.
Another might try looking in relation to the Temperley Lieb-Algebra or other Planar Algebras.