The study of bijections between combinatorial objects has always remained an attractive part of Enumerative and Algebraic Combinatorics. Despite their wide applicability, their remains a complete lack of formal understanding as to which bijections are equivalent, and which one are "good" in a sense that they preserve the natural structure of the combinatorial objects. We initiate the study of asymptotic properties of partition bijections and claim that a large class of well known bijections are in fact "asymptotically stable". Using CS-style reduction ideas we propose the first formal way to formulate that all classical Young tableau bijections are in fact linear tie equivalent. Other combinatorial objects and several new directions are also discussed.
Since the early days of mathematics, combinatorial objects have played an important role. These objects, which include various sets of graphs, trees, partitions of integers, tilings of regions with smaller shapes, etc. Finding or estimating the number of such objects is a fundamental problem which has been resolved in some instances and remains open in many other case. Sometimes one is able to relate the number of such objects to the number of other objects by means of a direct combinatorial argument (a bijection). We propose an in-depth study of the nature of these bijection, as to whether (and how) they reveal not just the number, but structural results on these combinatorial objects.
