Freely generated magmas and the Catalan family: a universal bijection
Seminar/Forum
Catalan numbers count a large number of very different sets of geometrical objects or numerical sequences. The first appearance was in the early 18th century and more are still being discovered (e.g. Catalan floorpans  2018). There are now over 200 such families. One method of proving a new set is Catalan is to biject it to a known Catalan family. This has resulted in a large number of such bijections (at least 200).
I will define a free magma generated by some finite set and present a theorem that can be used to prove when a magma is freely generated. The theorem can be used to prove a Catalan family is a free magma (generated by a single element). A simple magma isomorphism between all such Catalan magmas gives a universal Catalan bijection: the single bijection for any pair of families. This brings some order to the current adhoc collection of such bijections. The universal bijection requires the full factorisation of an object. To facilitate this the idea of an embedded bijection of a binary tree into the object is defined. This magnetisation of the Catalan family can be extended to others families such as the Motzkin and Schröder families.
Other information: This talk will be broadcast online using Zoom conferencing system. Join from PC, Mac, iOS or Android: https://unimelb.zoom.us/j/694937879
Presenter

Dr Richard Brak, The University of Melbourne