Málstofa: Giulio Cerbai – Stærðfræðistofa

Fyrirlesari: Giulio Cerbai, Raunvísindastofnun Háskóla Íslands

Titill: A combinatorial theory of transport of patterns

Staðsetning: Tg-227 í Tæknigarði
Tími: Fimmtudagur 24.febrúar kl. 10:30 / 3. mars kl. 10:30

Ágrip:

Combinatorics study enumerative, algebraic and geometric properties of families of discrete objects. Some of them can be equipped with a notion of pattern containment. The resulting posets have been extensively studied, but few results that link different structures are known. Our work aims to develop a unifying theory of transport of patterns capable of handling various families of objects. By revealing links between combinatorial structures and providing new enumerative results, this would improve our global understanding of pattern avoidance.

The first part of this seminar is devoted to a gentle introduction to combinatorics and, more specifically, permutation patterns. We provide the necessary background and define a hierarchy of endofunctions that embody the structures covered by our work. We then state the two fundamental aspects to transport of patterns: (i) defining suitable pattern-transporting maps between sets of endofunctions; (ii) understanding how such maps behave with respect to pattern containment. Finally we introduce the Burge transposition on biwords and show how this operation can be used to obtain a transport theorem for (modified) ascent sequences and Fishburn permutations.

In the second part of this seminar, we show numerous examples and applications of the proposed framework, as well as some related research directions. We also show how the Burge transpose can be used to provide a generalization of the Eulerian polynomial over Cayley permutations.