Enumerative combinative: interactions with algebra, number theory and physics – COMBINE

Enumerative combinatorics is the area of combinatorics that aims at finding the number of ways that certain patterns can be formed. The problem of discovering an enumeration formula frequently involves deriving a recurrence relation or generating function, and using this to arrive at the desired form. When the formula is complicated, a simple asymptotic approximation may be preferable.
Our goal is to develop general and efficient techniques in Enumerative Combinatorics
in order to attack problems coming from Combinatorics, Algebra, Number Theory and Physics.
One such example, is the proof of the Razumov--Stroganov conjecture
due to Cantini and Sportiello. This problem came from physics and was solved using beautiful
combinatorial arguments. Our objective is to continue to build enumerative combinatorics as a strong field
thanks to such an interdisciplinary approach. In particular when we use combinatorial arguments that give a bijection between two equinumerous states, we also want to understand the finer structure of the objects, their symmetries, and their refinements, in order to prove important results in other fields.

To make progress in these directions, we have access to a wide range of techniques.
One of the main tools of enumerative combinatorics is generating functions.
We can study them using their functional equations to understand their nature:
rational, algebraic, D-finite... Several tools coming from
asymptotic and analytic combinatorics, and computer algebra, can
be used to solve these problems.
We can also use number-theorical techniques to
study the nature of generating functions and prove some congruence results
for their coefficients. Coming from integrable systems and algebra, we master a lot of techniques to enrich the combinatorial models with parameters in order to make the proofs transparent. To prove combinatorial identities we can now use computer algebra techniques, but also sophisticated methods coming from hypergeometric series and symmetric functions.
The innovative nature of the project is that we are all enumerative combinatorialists at heart but we use different techniques to solve our combinatorial problems. We are planning to team up and work together to build the combinatorics of the future: a strong field with lots of deep techniques. We believe that this consortium, with its strengths in enumeration, physics, number theory and algebra, is the ideal team. We think that it is the right time for this project, as a number of us are now mature researchers.

Our proposal is organized around four main themes: Poset intervals, Combinatorics and statistical mechanics, Combinatorics of maps and walks and permutations, Enumerative problems coming from algebra. Nevertheless some questions on permutations are related to combinatorial objects coming from statistical mechanics, some questions about poset intervals
come from algebra... So these four themes are inteconnected and also interest participants in the three sites of
the proposal.

The consortium is interdisciplinary and multisite. The Paris site
is composed of: Guillaume Chapuy (CR, IRIF), Sylvie Corteel (PI, DR, IRIF),
Enrica Duchi (MDC, IRIF), Eric Fusy (CR, LIX), Matthieu Josuat-Vergès (CR, IGM),
Jeremy Lovejoy (CR, IRIF) and Andrea Sportiello (CR, LIPN). The Lyon site is composed of Riccardo Biagioli (MDC, ICJ), Jérémie Bouttier (CR, CEA), Frédéric Chapoton (DR, IRMA), Jehanne Dousse (CR, ICJ), Frédéric Jouhet (MDC, ICJ) and Philippe Nadeau (PI, CR, ICJ).
The Bordeaux site is composed of Jean-Christophe Aval (PI, CR, LABRI),
Adrien Boussicault (MDC, LABRI), Yvan Le Borgne (CR, LABRI) and Mireille Bousquet-Mélou (DR, LABRI).
The team is made 17 combinatorialists which are computer scientists (10), mathematicians (6) and physicists (1).