Schedule

Talks will take place in the Mike & Ophelia Lazaridis Quantum Nano Centre (QNC) (see map) in rooms QNC 1501 for registration/breaks and QNC 0101 for talks. Talks will be streamed. Please email one of the organizers to receive the link.

Program

Poster titles. For abstracts, see the page of the poster session.


  • A New modular symmetric function and its applications: Modular s-Stirling numbers, Bazeniar Abdelghafour, Moussa Ahmia, José L. Ramírez, Diego Villamizar (University of Mohamed Seddik Benyahia, Algeria)

  • Connective Constant of Honeycomb Lattice, Irha Ali (Waterloo)

  • On the average number of cycles in conjugacy class products, Jesse Campion Loth, Amarpreet Rattan (Simon Fraser)

  • Cyclic sieving for block factorizations of the long cycle, Justin Bailey and Theo Douvropoulos (UMass Amherst)

  • Faces of the generalized Pitman-Stanley polytope, William Dugan, Maura Hegarty, Alejandro Morales, and Annie Raymond (UMass Amherst)

  • Counting isospectral magnetic graphs, John Stewart Fabila-Carrasco (Waterloo):

  • Arithmetical structures on bidents, Kassie Archer, Abigail C. Bishop, Alexander Diaz-Lopez, Luis D. Garcia Puente, Darren Glass, Joel Louwsma (Waterloo):

  • Pattern avoidance and connectivity in chord diagrams, Ali Assem Mahmoud (Perimeter), Lukas Nabergall (Waterloo):

  • Universal Partial Cycles on De Bruijn Graphs, D. Fillmore, B. Goeckner, R. Kirsch, J. Martin (Iowa State University), D. McGinnis

  • Longest paths related to Steenrod length of real projective spaces, Khanh Duc Nguyen (University at Albany - State University of New York):

  • Naruse hook formula for linear extensions of mobile posets, GaYee Park (UMass Amherst):

  • A new order on integer partitions, Étienne Tétreault (LaCIM, UQAM):

  • A Subdivision Algebra for a Product of Two Simplices Via Flow Polytopes, Matias von Bell (University of Kentucky):




Talks, Some slides available below.

Three of the most classical and well-known identities in the theory of partitions concern: (1) the generating function for p(n) (Euler); (2) the generating function for partitions into distinct parts (Euler), and (3) the generating function for partitions in which parts differ by at least 2 (Rogers-Ramanujan). The lovely, simple argument used to produce the relevant generating functions is mostly never seen again. Actually, there is a very general theorem here which we shall present. We then apply it to prove two familiar theorems; (1) Goellnitz-Gordon, and (2) Schur 1926. We also consider an example where the series representation for the partitions in question is new. We close with an application to "partitions with n copies of n."


The last 30 years have seen increasing and interesting interactions between the combinatorics of generalized Dyck paths and parking functions, Macdonald polynomials and operators, diagonal coinvariant spaces, Hilbert schemes of points, Khovanov-Rozansky homology of $(m,n)$-torus links, Double affine Hecke algebra, Elliptic Hall algebra, and Superspaces of bosons and fermions. The aim of this talk is to describe a broad extensions of the combinatorics involved in these contexts.



We discuss the use of experimental mathematics in a pedagogical context, with mathematics majors in a computation class. We'll show how the class was led to discover a beautiful fact about Newton Raphson iterations on quadratic functions.


  • A log/tropical take on double Hurwitz numbers, Renzo Cavalieri (Colorado State University):

After a review of some classical results about double Hurwitz numbers, I will present some recent joint work with Hannah Markwig and Dhruv Ranganathan, in which we interpret double Hurwitz numbers as intersection numbers of the double ramification cycle with a logarithmic boundary class on the moduli space of curves. This approach removes the "need" for a branch morphism and therefore allows the generalization to related enumerative problems on moduli spaces of pluricanonical divisors - which have a natural combinatorial structure coming from their tropical interpretation.



Over 25 years ago Goulden and Jackson posed a mysterious conjecture which connects a generating function of Jack symmetric functions with enumeration of combinatorial maps. We describe the necessary background of this intriguing conjecture, and discuss the progress made over the passing years towards its resolution. Finally, we show how Goulden and Jackson's insight inspired us to introduce the $b$-deformed weighted Hurwitz numbers and we describe their positivity (joint work with Guillaume Chapuy) and applications in matrix models (joint work with Valentin Bonzom and the aforementioned author).



We discuss two problems regarding the refined enumeration of lattice paths in the plane with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the descents), and the number of crossings. In the first case, we count single paths by the number of times they cross a fixed line; in the second, we count pairs of paths by the number of times they cross each other.

All of our proofs are bijective, and the answers are given by remarkably simple formulas involving q-binomial coefficients.


A meandric system of size n is a non-intersecting collection of closed loops in the plane crossing the real line in exactly 2n points (up to continuous deformation). Connected meandric systems are called meanders, and their enumeration is a notorious hard problem in enumerative combinatorics. In this talk, we discuss a different question, raised independently by Goulden-Nica-Puder and Kargin: what is the number of connected components cc(M_n) of a uniform random meandric system of size 2n? We prove that this number grows linear with n, and concentrates around its mean value, in the sense that cc(M_n)/n converges in probability to a constant.

Our main tool is the definition of a notion of local convergence for meandric systems, and the identification of the “quenched Benjamini-Schramm” limit of M_n. The latter is the so-called infinite noodle, a 1D (but difficult to study!) percolation model recently introduced by Curien, Kozma, Sidoravicius and Tournier. Our main result has also a geometric interpretation, regarding the Hasse diagram H_n of the non-crossing partition lattice NC(n): informally, our result implies that, in H_n, almost all

pairs of vertices are asymptotically at the same distance from each other. We use here a connection between H_n and meandric systems discovered by Goulden, Nica and Puder.

Based on joint work with Paul Thevenin, arXiv:2201.11572.



Hamel-Goulden determinantal identities have recently been shown by Foley and King to apply to a tableau-based version of Macdonald's ninth variation skew Schur functions. In this talk we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. At the level of the ninth variation the corresponding determinantal identities are all distinct, but the original notion of supersymmetry is lost. We show this can be remedied at the level of Macdonald's sixth variation involving a doubly infinite sequence of factorial parameters. This is joint work with R.C. King.


Let A be an alphabet and let F be a set of words with letters in A. With the help of the Goulden-Jackson cluster theorem, we show that the sum of all words with letters in A with no consecutive subwords in F, as a formal power series in noncommuting variables, is the reciprocal of a series with all coefficients 0, 1 or −1. We also explain how this result is related to a theorem of Curtis Greene on lattices with Möbius function 0, 1, or −1 and to the work of Dotsenko and Khoroshkin.



A number of sequences of combinatorial interest are the sequences of moments of a probability density. A probability density determines an inner product on polynomials, and hence a sequence of orthogonal polynomials. To each sequence of orthogonal polynomials, there is an associated three-term recurrence or, equivalently, an infinite weighted path. The closed walks on these paths provide a combinatorial view of the associated moments and polynomials. I will discuss these connections, and also indicate how we can derive information about properties of certain quantum walks.


  • Early Work, Ian Goulden (Waterloo): I will talk briefly and informally about my work in the last century.



  • Divided Difference Operators for Hessenberg Varieties, Mathieu Guay-Paquet (LaCIM UQAM):

Classically, Schubert polynomials form a basis for the cohomology ring of the flag variety, and they can be defined using divided difference operators.

In this talk, we generalize the divided difference operators to the larger cohomology rings of regular semisimple Hessenberg varieties (a family of Catalan-many sub-varieties of the flag variety). With these generalized operators, we can decompose the Hessenberg cohomology rings in a way which categorifies the modular relation of chromatic quasi-symmetric functions.


Let $T$ be a noncrossing tree on vertices $\{1,2,\ldots,n\}$. We say $T$ is \emph{locally oriented} if each edge is directed toward its smallest endpoint. Let $e_i(T)$ (respectively, $d_(T)$) be the number of vertices of indegree (outdegree) $i$ in $T$. Okoth and Wagner found an elegant multiplicative formula counting locally-oriented noncrossing trees with specified in- and out-degree vectors $(e_i(T))_{i \geq 0}$ and $(d_i(T))_{i \geq 0}$. We generalize their result to give a simple formula counting these trees with specified numbers $(n_{ij})_{i,j \geq 0}$ of vertices of indegree $i$ and outdegree $j$.


  • At It and In It, David Jackson (Waterloo):

The Early Days, and a Combinatorialist’s Retrospection in Seven Themes.


The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length ("genus 0"). We study the analogue of these numbers for reflection groups, with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group W is full if the factors generate the whole group W. For the combinatorial families, we compute the generating function for full factorizations of arbitrary length for arbitrary elements, in terms of the generating functions of the symmetric group and an appropriate cyclic group. As a corollary, we obtain leading-term formulas which count minimum-length full reflection factorizations of arbitrary elements in terms of the Hurwitz numbers of genus $0$ and $1$ and number-theoretic functions. In the case of an arbitrary complex reflection group, we give uniform (i.e., case-free) formulas for the number of genus-0 full reflection factorizations of a large family of elements (the parabolic quasi-Coxeter elements).


  • Graphs in surfaces, their one-face subgraphs, and the critical group, Iain Moffatt (Royal Holloway, University of London):

Critical groups are groups associated with graphs. They are well-established in combinatorics; closely related to the graph Laplacian and arising in several contexts such as chip firing and parking functions. The critical group of a graph is finite and Abelian, and its order is the number of spanning trees in the graph, a fact equivalent to Kirchhoff’s Matrix--Tree Theorem.


What happens if we want to define critical groups for graphs embedded in surfaces, rather than for graphs in the abstract?

In this talk I'll offer an answer to this question. I'll describe an analogue of the critical group for an embedded graph. We'll see how it relates to the classical critical groups, as well as to Chmutov's partial-duals, Bouchet's delta-matroids, and a Matrix--quasi-Tree Theorem of Macris and Pule.

This is joint work with Criel Merino and Steven D. Noble.


Some people swear by "Gödel, Escher, Bach" as a source of inspiration, but I prefer Hurwitz, Goulden, Jackson. Hurwitz's 1891 study of branched covers of the sphere, in which both the exponential formula and the braid groups first appeared, is a watershed work in enumerative and algebraic combinatorics. However, it was Goulden and Jackson's 1997 rediscovery and proof of Hurwitz's putative counting formula for genus zero covers that both laid the foundations and provided the impetus for Hurwitz theory to grow and flourish into one of the most fruitful research topics in contemporary combinatorics. I will pay tribute to Goulden and Jackson's vision by describing how it ultimately provided exactly the right perspective on the Brezin-Gross-Witten and Harish-Chandra/Itzykson-Zuber integrals, two (in)famous matrix integrals whose fine structure is enormously important in mathematical physics but would have otherwise remained unknown.


One of the most remarkable formulas in Algebraic Combinatorics is the hook-length formula for the number of Standard Young Tableaux. While no such nice product formula exists for the number of skew SYTs in general, since 2014 there is a formula which gives the number of SYTs of a given skew shape as a sum over certain ("excited") diagrams of product of their hook lengths. In this talk I'll show generalizations of this formula and several proofs, one of which uses the determinantal formula of Goulden and Hamel for skew Schur functions.



The Murnaghan–Nakayama rule is a semi-combinatorial formula for computing characters of irreducible symmetric group representations $\chi^\lambda(\sigma)$, $\sigma \in S_n$. The formula is "combinatorial" in the sense that it involves counting certain tableaux, but only "semi" because these tableaux must be counted with signs. The more one examines the formula, the more curious it seems. What does it all really mean?

In a partial attempt to answer this question, consider the special case where the order of $\sigma$ is an odd prime power. In this case, I will describe a geometric interpretation of the Murnaghan–Nakayama rule in terms of certain (non-transverse) Schubert intersections. This talk is based on joint work with Jake Levinson.


  • Goulden and Yong's dual construction on factorizations of the full cycle and Mahonian statistics on trees, Amarpreet Rattan (Simon Fraser University):

A factorization of the canonical full cycle into transpositions can be represented naturally as a tree. Goulden and Yong (2002) showed that taking a dual of this tree produces a bijection from these factorizations to vertex labelled trees that map natural statistics on factorization to natural statistics on trees. In recent work with Irving, we use a similar map to show that a Mahonian statistic on trees, the major index, has a simple interpretation for factorizations. One advantage of this approach is that it easily generalizes to a correspondence between factorizations of the canonical full cycle into k-cycles and cactii which preserve more general Mahonian statistics.


In this talk, I will present two bijective proofs of generalizations of Catalan numbers involving Dyck paths. The first one is the generalized ballot problem of counting Dyck paths that never go below the line with slope k (work with I. P. Goulden). The second one is a bijection between k-triangulations and k-fans of Dyck, paths, counted by a determinant of Catalan numbers.


Let X be a subset of {(i,j) : 1\leq i,j \leq n, i\neq j}. The *X-descent set* of a permutation w = a_1 ... a_n \in S_n is defined by XDes(w) = {i : (a_i,a_{i+1})\in X}. If X = {(i,j) : n\geq i>j\geq 1}, then XDes(w) = Des(w), the ordinary descent set. We define a quasisymmetric function U_X which is a generating function for permutations w\in S_n according to their X-descent set. It turns out that U_X is a symmetric function whose properties we will discuss. We also discuss some connections with Hamiltonian paths in digraphs.


Marko Thiel found a sequence of combinatorial objects $X_n$, each carrying a natural action of the cyclic group $C_{n-1}$ of order $n-1$ such that the triple $\left(X_n,C_{n-1},\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q\right)$ exhibits the cyclic sieving phenomenon. He called these objects $(1,2)$-configurations and asked if there was a natural statistic on this set whose generating function is the $q$-Catalan polynomial $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. We describe such a statistic.


The structure constants of the significant subalgebras of the group algebra of the symmetric group and its generalisations are salient combinatorial objects. On the one hand, they give the number of factorisations of a given permutation into a product of permutations with given statistics. On the other hand, they are involved in various important relations for symmetric or quasisymmetric functions. The works of I. Goulden and D. Jackson give inspiring formulas and conjectures regarding the structure constants of the class algebra and the double coset algebra. Furthermore, using the theory of Jack symmetric functions they provide a major conjecture regarding some interpolation between the two families of constants. Instead of its conjugacy class, one may look at the descent or the peak set of a permutation. The structure constants of the descent algebra (the peak algebra) appear in factorisation formulas for I. Gessel’s fundamental (J. Stembridge’s peak) quasisymmetric functions. While very similar, the formulas of Gessel and Stembridge do not seem directly related. We introduce a q-deformation for the generating function of enriched P-partitions and provide a new interpolation between these two classical families of quasisymmetric functions. This new framework provides a link between the two formulas.


  • Algebraic underpinnings for quantum field theory, Karen Yeats (Waterloo) video:

Drawing on work of Jackson and collaborators as well as my own work and that of others, I will describe how the tools of algebraic combinatorics and algebraic enumeration are being used to build new underpinnings for quantum field theory.


Flow polytopes are polytopes defined by flows on directed acyclic graphs. One method to obtain unimodular triangulations of these polytopes is through a framing of the graph. We present results on framed triangulations of two families of flow polytopes. 1) Given a lattice path \nu, we study a family of flow polytopes whose normalized volumes are \nu-Catalan numbers, and show that these polytopes possess interesting triangulations whose dual graphs are the \nu-Tamari lattice, and the principal order ideal in Young's lattice generated by \nu. 2) For the class of graphs which possess ample framings, we show that top-dimensional simplices of an amply framed triangulation of the polytope correspond to support \tau-tilting modules of an associated path algebra over a quiver. In both cases, the poset structure of the dual of the triangulation allows us to obtain results regarding the h^*-vector of the polytopes.