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Martin Willert: Three Families Indexed by Catalan Objects

BSc thesis presentation

Time: Fri 2020-06-05 09.30 - 10.30

Location: Zoom, meeting ID: 635 5708 7798

Participating: Martin Willert

Supervisor: Per Alexandersson


Within the field of algebraic combinatorics, there has been some recent interest in the bijective correspondence between families indexed by Catalan objects. In this thesis, we look at three such families: acyclic orientations, rook placements and perfect matchings.

We search for bijections between these three families, and we use q-analogs to refine our results. We are particularly interested in bijections which behave well with respect to such q-analogs. We find that these families are in bijective correspondence, and we propose using this correspondence to assist future advances in some well-known open problems of the field, for example the Stanley–Stembridge Conjecture.