Mardi 21 Octobre
Heure: 
14:00  17:00 
Lieu: 
Salle B107, bâtiment B, Université de Villetaneuse 
Résumé: 
A general theory of Wilfequivalence for Catalan structures 
Description: 
Mathilde Bouvel It is a commonly observed phenomenon in enumerative combinatorics thatseveral combinatorial classes share the same enumeration. For any twoclasses which seem to have the same enumeration sequence, a naturalproblem is to prove that it is indeed the case, ideally with abijective proof that allows to map the structure of one class to thatof the other.Such coincidences of enumeration are called Wilfequivalences in thecontext of patternavoiding permutation classes (the definition ofpatternavoidance will be given during the talk). Wilfequivalence hasbeen a popular topic in the research on patternavoiding permutations,from its beginnings in the seventies until now. It is fair to say thatmost of the works done so far are specific to given pairs ofequinumerous classes, thus forming a sort of "casebycase catalogue"of the known Wilfequivalences.In this talk, we explore a different approach: we are interested indescribing all Wilfequivalences between permutations classes definedby the avoidance of two patterns: 231 and an additional pattern p(w.l.o.g., we can assume that p itself avoids 231). We will explainthat this is one way of phrasing a seemingly more general (butactually equivalent) question: that of describing allWilfequivalences between classes of Catalan objects subject to onefurther avoidance restriction. Such classes are denoted Av(A), A beinga Catalan object.Our results, to be presented in the talk, are the following.First, we define an equivalence relation ~ among Catalan objects. Ourmain result is that it refines Wilfequivalence: if A ~ B, then Av(A)and Av(B) have the same enumeration. The proof is subdivided inseveral cases, and it is bijective in all cases but one. We furtherconjecture that the converse statement holds, i.e., that this relation~ coincides with Wilfequivalence.Then, we show how to enumerate the number of equivalence classes for~, hereby providing an upper bound on the number of Wilfequivalenceclasses.It is also interesting to study a special ~equivalence class, whichcan be understood at a very fine level of details. Our results on this~class (called the "main" one) unify and generalize several resultsof the literature on Wilfequivalence.Finally, we explain how our bijective cases in the proof of the maintheorem can often be refined to provide comparison results between theenumerations of classes Av(A) and Av(B) when A and B are notequivalent for ~. A consequence is that the "main" ~class correspondsto the largest possible enumeration sequence.This is a joint work with Michael Albert (University of Otago), and apreprint is available at arXiv:1407.8261. 

