Freely generated magmas and the Catalan family: a universal bijection
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 ad-hoc 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
Dr Richard Brak, The University of Melbourne