From geometry to generating functions: rectangulations and permutations

Abstract: A rectangulation of size n is a tiling of a rectangle by n rectangles such that no four rectangles meet in a point. In the literature, rectangulations are also called floorplans or rectangular dissections. In this talk, we will analyse several classes of pattern-avoiding rectangulations which lead to surprisingly nice enumerative results and new bijective links with pattern-avoiding permutations. We prove that their generating functions are algebraic, and confirm several conjectures by Merino and Mütze. We also analyse a new class of rectangulations, called whirls: they are related to Catalan numbers, but no simple proof of it is known! We prove this fact using a generating tree. This leads to an intricate functional equation, for which the method of resolution has its own interest.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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