*Résumé :* It is a commonly observed phenomenon in enumerative combinatorics that
several combinatorial classes share the same enumeration. For any two
classes which seem to have the same enumeration sequence, a natural
problem is to prove that it is indeed the case, ideally with a
bijective proof that allows to map the structure of one class to that
of the other.
Such coincidences of enumeration are called Wilf-equivalences in the
context of pattern-avoiding permutation classes (the definition of
pattern-avoidance will be given during the talk). Wilf-equivalence has
been a popular topic in the research on pattern-avoiding permutations,
from its beginnings in the seventies until now. It is fair to say that
most of the works done so far are specific to given pairs of
equi-numerous classes, thus forming a sort of "case-by-case catalogue"
of the known Wilf-equivalences.
In this talk, we explore a different approach: we are interested in
describing all Wilf-equivalences between permutations classes defined
by 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 explain
that this is one way of phrasing a seemingly more general (but
actually equivalent) question: that of describing all
Wilf-equivalences between classes of Catalan objects subject to one
further avoidance restriction. Such classes are denoted Av(A), A being
a Catalan object.
Our results, to be presented in the talk, are the following.
First, we define an equivalence relation ~ among Catalan objects. Our
main result is that it refines Wilf-equivalence: if A ~ B, then Av(A)
and Av(B) have the same enumeration. The proof is subdivided in
several cases, and it is bijective in all cases but one. We further
conjecture that the converse statement holds, i.e., that this relation
~ coincides with Wilf-equivalence.
Then, we show how to enumerate the number of equivalence classes for
~, hereby providing an upper bound on the number of Wilf-equivalence
classes.
It is also interesting to study a special ~-equivalence class, which
can be understood at a very fine level of details. Our results on this
~-class (called the "main" one) unify and generalize several results
of the literature on Wilf-equivalence.
Finally, we explain how our bijective cases in the proof of the main
theorem can often be refined to provide comparison results between the
enumerations of classes Av(A) and Av(B) when A and B are not
equivalent for ~. A consequence is that the "main" ~-class corresponds
to the largest possible enumeration sequence.
This is joint work with Michael Albert (University of Otago), and a
preprint is available at arXiv:1407.8261.

[arXiv]

Dernière modification : Wednesday 12 October 2022 |
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