François Bergeron (Université du Québec à Montréal, Canada)
FPSAC: Flajolet, Power Series and Analytic Combinatorics
Ron Graham (University of California, San Diego, USA)
Generating functions for restricted Eulerian numbers
Abstract: In this talk I will describe some recent work with Fan Chung on certain joint statistics for permutations π ∈ Sn. These involve the number of descents of π, the maximum drop of π and the value of π(n), and result in some new identities for restricted Eulerian numbers.
0-Hecke algebra actions on coinvariants and flags
Ilse Fischer (University of Vienna, Austria)
An alternative approach to alternating sign matrices
Abstract: Alternating sign matrices were first defined by Robbins and Rumsey in the early 1980s when they discovered that the λ-determinant, a natural generalization of the determinant, has an expansion as a sum over all alternating sign matrices, just as the ordinary determinant has an expansion as a sum over permutation matrices. Later it was observed that physicists had been studying a model for square ice that is equivalent to alternating sign matrices for a long time. Since then these square ice techniques are a standard tool to attack various enumeration problems related to alternating sign matrices.
In my talk I shall present an alternative approach to alternating sign matrix enumeration which is more in the spirit of Zeilberger's original proof of the alternating sign matrix theorem. Starting point is an operator formula for the number of monotone triangles with prescribed bottom row. Refined enumerations of alternating sign matrices with respect to a fixed set of boundary columns and rows can be expressed in terms of this operator formula. This enables us to translate certain identities for the operator formula to identities for refined enumerations of alternating sign matrices. This leads, on the one hand, to systems of linear equations that determine the numbers uniquely, and, on the other hand, to surprisingly simple linear relations between them. I will also report on recent attempts to translate these calculations into more combinatorial reasonings.
Joel Lewis, Ricky Liu, Alejandro Morales, Greta Panova, Steven Sam and Yan Zhang
Matrices with restricted entries and q-analogues of permutations
Allowed patterns of beta-shifts
Jean-Christophe Aval, Adrien Boussicault and Philippe Nadeau
Vincent Pilaud and Francisco Santos
The brick polytope of a sorting network
Lauren Williams (University of California, Berkeley, USA)
KP solitons, total positivity, and cluster algebras
Abstract: Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. More recently, several authors have focused on understanding the regular soliton solutions that one obtains in this way: these come from points of the totally non-negative Grassmannian.
In joint work with Yuji Kodama, we establish a tight connection between Postnikov's theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation. This connection allows us to apply machinery from total positivity to KP solitons. In particular, we classify the soliton graphs coming from the totally non-negative Grassmannian, when the absolute value of the time parameter is sufficiently large. We demonstrate an intriguing connection between soliton graphs and the cluster algebras of Fomin and Zelevinsky. Finally, we apply this connection towards the inverse problem for KP solitons.
Takeshi Ikeda, Yasuhide Numata and Hiroshi Naruse
Bumping algorithm for set-valued shifted tableaux
Tableaux and plane partitions of truncated shapes
Sara Billey and Andrew Crites
Rational smoothness and affine Schubert varieties of type A
Richard Ehrenborg (University of Kentucky, USA)
The Law of Aboav--Weaire and its analogue in three dimensions
Abstract: When investigating the structure of metals it is known that the atoms lie in a lattice structure. However, the lattice property only holds locally, that is, in a three dimensional cell called a grain. Bordering the grain is a boundary where the atoms lie chaotically, and beyond that is a new grain where the lattice has a different orientation. The structure of these grains amounts to a three dimensional simple subdivision of space.
Looking at the two dimensional analogue, one observes that grains with a small number of sides tend to be surrounded by grains with a large number of sides, and vice versa. The Law of Aboav--Weaire states that the average number of sides of the neighbors of an n-sided grain should be roughly 5+6/n. By introducing the correct error term we prove this law of Material Science and discuss its extension to three dimensions.
This is joint work with Menachem Lazar and Jeremy Mason. Moreover, selected work of von Neumann, MacPherson and Srolovitz will be presented.
Maciej Dołęga and Piotr Śniady
Polynomial functions on Young diagrams arising from bipartite graphs
Hyperplane arrangements and diagonal harmonics
A polynomial expression for the Hilbert series of the space of diagonal harmonics
Matthieu Josuat-Vergès and Jang Soo Kim
Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity
Monica Vazirani (University of California at Davis, USA)
Simple KLR modules
Abstract: Khovanov-Lauda-Rouquier (KLR) algebras have played a fundamental role in categorifying quantum groups. I will discuss the structure of their simple modules, in particular that they carry the structure of a crystal graph. This is joint work with Aaron Lauda.
Drew Armstrong and Brendon Rhoades
The Shi arrangement and the Ish arrangement
Stefan Forcey, Aaron Lauve and Frank Sottile
Cofree compositions of coalgebras
Marcelo Aguiar and 25 co-authors
Supercharacters, symmetric functions in noncommuting variables, extended abstract
Excursion & banquet
Marc Noy (Polytechnic University of Catalonia, Spain)
Counting maps and graphs
Abstract: The theory of map enumeration was started by Tutte in the 1960s, in an attempt to shed light on the four colour problem, and since then the field has grown considerably. Many classes of maps have been enumerated, including maps on surfaces, and connections have been found to other fields, particularly to statistical physics. More recently, several classes of (unembedded) graphs have been analyzed, in particular planar graphs and graphs on surfaces, and precise asymptotic estimates have been obtained. In the talk we will review these results and the companion results on the structure of random graphs from these families. The main tool in our work is analytic combinatorics, as developed by Philippe Flajolet.
Andrew Goodall, Criel Merino, Anna de Mier and Marc Noy
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1)
Alex Fink and David Speyer
K-classes for matroids and equivariant localization
The equivariant topology of stable Kneser graphs
Lior Pachter (University of California, Berkeley, USA)
Affine and projective tree metric theorems with applications to phylogenetic reconstruction
Abstract: The tree metric theorem provides a combinatorial four point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A similar (but weaker) four point condition characterizes dissimilarity maps derived from circular split systems (Kalmanson metrics). The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We will review these theorems via the unifying framework of the (tropical) space of trees and extensions to PQ- and PC-trees, and will discuss applications to phylogenetic reconstruction. In particular, we will provide an explanation for the peculiar form of the balanced minimum evolution criterion popular in phylogenetics. The work to be presented is joint with Aaron Kleinman.
Gregg Musiker and Victor Reiner
A topological interpretation of the cyclotomic polynomial
Submaximal factorisations of a Coxeter element in complex reflection groups
Pavle V. M. Blagojevic, Benjamin Matschke and Günter M. Ziegler
A tight colored Tverberg theorem for maps to manifolds
Richard Kenyon (Brown University, USA)
The octahedral recurrence and generalizations
Abstract: This is joint work with A. Goncharov. The octahedral recurrence, or Hirota equation, is a well-known integrable discrete dynamical system, related to alternating sign matrices and domino tilings of Aztec diamonds.
We show that there is an underlying completely integrable Hamiltonian system commuting with the octahedral recurrence. Formulas relating it to the octahedral recurrence can be written explicitly in terms of dimers.
Similar systems exist for any periodic planar graph.
Jason Bandlow, Anne Schilling and Mike Zabrocki
The Murnaghan-Nakayama rule for k-Schur functions
Masato Okado and Reiho Sakamoto
Stable rigged configurations and Littlewood-Richardson tableaux
Kerov's central limit theorem for Schur-Weyl and Gelfand measures
Stefan Felsner (Technical University of Berlin, Germany)
Abstract: A squaring is a tiling into squares of different sizes. In a seminal paper Brooks, Smith, Stone and Tutte (1940) discussed squarings related to segment contact representations of planar quadrangulations. Regarding the squares of a squaring as vertices and edges as being defined by contacts we obtain the square dual graph. Schramm (1993) showed that 5-connected inner triangulations of a 4-gon can be represented as square duals. In this talk we review the plane situation and present some results concerning squarings of the torus and the graphs represented by them.
(joint work with E. Fusy)
The pentagram map and Y-patterns
Valentin Féray and Piotr Śniady
Dual combinatorics of zonal polynomials
Art Duval, Caroline Klivans and Jeremy Martin
Critical groups of simplicial complexes