Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this talk is to show that under certain positivity assumptions the dominant singularity of the solution function of a system of linear catalytic equations has a universal square-root singularity. Actually it is possible to define a so-called grammar which leads to a positive system of polynomial equations (without a catalytic variable) so that general theorems can be applied. This is joint work with Cyril Banderier.

In 2020, Bonnet et al. invented the graph invariant "twin-width" relying on contraction sequences for graphs and showed that on graphs of bounded twin width many classical graph algorithms become tractable. In 2025, Ahn et al. showed that twin-width 1 graphs are permutation graphs and gave an algorithm to compute the contraction sequence efficiently, which follows closely the permutation diagram. We do a reverse-engineering of this algorithm to generate permutation diagrams that correspond to twin-width 1 graphs at least once. This gives an upper bound of the form $c^n n!$ for the number of twin-width 1 graphs with explicit $c$. Joint work with Zbigniew Golebiewski, Malgorzata Sulkowska and Michael Wallner.

Sums involving binomial coefficients and other hypergeometric expressions occur frequently in enumerative and analytic combinatorics. There is a well-established mathematical toolkit for dealing with these sums, based on techniques such as the (discrete) Laplace method, the Stirling approximation of factorials and binomial coefficients, and the Mellin transform. In this talk, we illustrate how a newly developed package for the computer mathematics system SageMath can be utilized to perform a full asymptotic analysis in the framework above – including explicit error bounds for the determined asymptotic behavior. We use the package to first treat a classical problem due to Ramanujan, and then answer a question due to Bóna and DeJonge concerning 132-avoiding permutations with a unique longest increasing subsequence.

A rooted planar map is a planar graph embedded on the sphere with one oriented edge. For a fixed rooted planar map P, we want to know how often the map P occurs as a submap in a random rooted planar map. In 1992, Bender, Gao and Richmond proved that the number of submap occurrences of P is almost surely linear, however the distribution of the submap counts remained a conjecture. We present some recent work with Michael Drmota and Nick Wormald where we prove a central limit theorem for such submap counts.

As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labeled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. In this talk, I will define random recursive metric spaces, and outline the proofs of a law of large numbers and a central limit theorem for the insertion depth; the distance from the master hook to the latch chosen. I will discuss a classic argument for random recursive trees which shows that the insertion depth is distributed as a sum of Bernoulli random variables. This argument is generalized for random recursive metric spaces via a martingale central limit theorem.

Given a cube in the lattice $\mathbb{Z}^3$, we consider random walks from $(0,0,0)$ to $(n,n,n)$ that take only unit steps of type $(1,0,0)$, $(0,1,0)$, or $(0,0,1)$. We study the number of times these walks hit the main diagonal $(m,m,m)$ for $0 < m \leq n$, as well as the sizes of the corresponding excursions. This setting can be analyzed through the composition scheme $V(W(z)) = \frac{1}{1 - W(z)}$, viewing each walk as a sequence of excursions between diagonal hits. In fact, it is possible to show that the point process of diagonal hits converges in a more general setting where the composition scheme takes the form $V(W(z)) = \left( \sum v_n W(z)^n \right)^\alpha$, with $v_n$ slowly varying coefficients. This problem has been first studied by Kuban, Panholzer, Stufler and later generalized further in collaboration with Stufler.

Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and $b$. The $V$-edge substitution of $X$ is the graph $X[V]$ where each edge $[x,y] \in E_X$ is replaced by a copy of $V$, identifying $x$ with $a$ and $y$ with $b$ or vice versa. (The latter choice does not matter; it yields isomorphic graphs). The aim is to describe the spectrum of $X[V]$ in terms of the spectra of $X$ and $V$. Instead of the (equally tractable) spectra of the adjacency matrices, we consider the versions which are normalised by dividing each row of the latter by the row sum (the vertex degree). These are stochastic, reversible matrices, and our approach applies more generally to reversible transition matrices corresponding to arbitrary positive edge weights invariant under $\gamma$. We write $P$ for the transition matrix over $X$ and $Q$ for the one over $V$. Together, they induce the matrix $P_*$ over $X[V]$. There is not a nice and compact formula which says how to transform the spectra of $P$ and $Q$ into the spectrum of $P_*\,$. The results depend on issues like whether $X$ is bipartite (with more satisfactory results) and on the eigenvalues of the restriction of $Q$ to $V \setminus \{ a,b\}$, which need to be classified into 4 possible types, each to be handled differently. Also quite subtle is the issue of determining the multiplicities of the eigenvalues of $P_*$ in terms of the input. This is ongoing work with Thomas Hirschler (TU Graz). There are various interesting issues for future research.

Let $F$ be a graph on $r$ vertices and let $G$ be a graph on $n$ vertices. An $F$-factor in $G$ is a subgraph of $G$ composed of $n/r$ vertex-disjoint copies of $F$, if $r$ divides $n$. For instance, a $K_2$-factor is simply a perfect matching. The study of threshold functions for $F$-factors in $G(n,p)$ goes back to Erdős and Rényi themselves; but for general $F$ it was not until the 2008 breakthrough paper by Johansson, Kahn and Vu that the weak threshold for strictly 1-balanced $F$ was established. More recently, Riordan and Heckel obtained sharp thresholds for $F=K_r$ and so-called nice graphs, using sophisticated coupling arguments that utilize Kahn's recent celebrated solution of Shamir's problem on hypergraph matchings. This talk reviews these results and extends them to sharp thresholds for any strictly 1-balanced $F$. In particular, this confirms the thirty year old conjecture by Ruciński that the sharp threshold for the emergence of an $F$-factor coincides with the sharp threshold for the disappearance of the last vertex that is not contained in a copy of $F$. These results were obtained in joint work with A. Heckel, M. Kaufmann, N. Müller, and M. Pasch.