Random Walks in Restricted Domains and Standard Young Tableaux

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing in September 2021.

This project will be supervised by Prof Thomas Prellberg.

The aim of this project is to study the connections between Standard Young Tableaux (SYT) and Random Walks using enumerative and bijective methods. Combinatorial interpretations of SYT of bounded height have recently received much attention; a good survey of this active area in combinatorics is given in [1].

SYT of bounded height can both be represented using specific random walks in the positive orthant of Zd and a certain type of coloured Motzkin paths. This connection is understood well, and explicit bijections have been constructed [2].

In dimensions d=2 and d=3 there is a well-known connection with Dyck and Motzkin paths. Intriguingly, recent enumerative work [3] found that restricting the domain for the random walks is in some sense equivalent to considering lattice paths in a finite strip. This result is based on equinumeracy shown by comparing counting formulas, and no bijective proof is known. Also, the relevant restriction for SYT is not known.

There are two open problems to be worked on here. The first one is identifying the appropriate statistics for SYT and subsequently the provision of a bijective proof of the equinumeracy. The second one is to generalise these results to higher dimensions, and hence to SYT of arbitrary bounded height.

The application procedure is described on the School website. For further inquiries please contact Professor Thomas Prellberg at t.prellberg@qmul.ac.uk. This project is eligible for full funding, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process.