Résumé : Various combinatorial structures admit, in a broad sense,
a notion of irreducibility: graphs can be connected, permutations
can be indecomposable, polynomials can be irreducible, etc. We are
interested in the probability that any such labeled object picked
randomly is irreducible, as its size tends to infinity. In this
talk, for certain classes, we obtain the asymptotics for this
probability in a common manner. We show that the coefficients
appearing in those asymptotics are integers and can be interpreted
as the counting sequences of other “derivative”
structures. Moreover, we obtain asymptotic probabilities that a
random combinatorial object has a given number of irreducible
components. Applications include connected graphs, indecomposable
permutations, irreducible tournaments, connected square-tiled
surfaces, indecomposable perfect matchings, combinatorial maps,
etc. Also, using species theory, we treat the Erd ̋os–R ́enyi G(n, p)
model. This is a joint work with Thierry Monteil.
Bio : Khaydar Nurligareev is a research and teaching assistant (ATER) at
the LIPN, University Sorbonne Paris Nord, where he received a PhD
degree on October 2022. The main research of Khaydar is focused on
enumerative and analytic combinatorics, but his research interests
include other domains like probability theory, lattice models and
tilings as well.
Webpage: https://lipn.univ-paris13.fr/∼nurligareev/
