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This talk explores rational generalizations of classic combinatorial objects, focusing on noncrossing partitions and associahedra (polygon triangulations). Learn about rational Catalan numbers Cat(a,b) = (a+b choose a) / (a+b) for coprime integers a and b, which extend the classical Catalan numbers Cat(n,n+1) and Fuss-Catalan numbers Cat(n,mn+1). Discover how rational Dyck paths function as lattice paths in an a×b rectangle that stay above the diagonal. The presentation covers collaborative research with Brendon Rhoades and Nathan Williams on rational noncrossing partitions and associahedra, as well as work with Nick Loehr and Greg Warrington on rational generalizations of parking functions counted by b^{a-1}. This 37-minute lecture was delivered as part of the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability" at the Erwin Schrödinger International Institute for Mathematics and Physics.
