Conference Program

We prove a theorem which adds a new member to the famous Göellnitz-Gordon identities. We construct a "new system of recurrence formulas" in order to prove it. Presentation based on the article arXiv 2403.10189. slides

A canon permutation is a multiset permutation constructed from a pair: a Dyck path and a permutation. This is done by labeling the up-steps and down-steps using the permutation and then reading the Dyck path. We study descents in canon permutations, continuing a line of work initiated by Elizalde.
By fixing a Dyck path and varying over all permutations, we associate a descent polynomial to the Dyck path. We study these polynomials and find relations to known statistics on Dyck paths. In particular, the degree of this polynomial is related to peaks in the bounce path. Talk based on a joint work with Danai Deligeorgaki: arXiv 2603.22565. slides

We introduce nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference from classical walks is that its nondeterministic steps consist of sets of steps from a predefined set, allowing for parallel exploration of all possible extensions. We discuss in detail two particular nondeterministic step sets inspired by Dyck and Motzkin walks and show that several nondeterministic classes of lattice paths, such as nondeterministic bridges, excursions and meanders, are algebraic. We extend our results to general step sets, proving that nondeterministic bridges and several subclasses of nondeterministic meanders are always algebraic, but leaving the case of excursions as a conjecture.
This research is motivated by the study of networks involving encapsulation and decapsulation of protocols. Our results are obtained using generating functions, analytic combinatorics and additive combinatorics. Talk based on a joint work with Michael Wallner: arXiv 2311.03234. slides

Okada and Stembridge's Pfaffian formula for the enumeration of families of nonintersecting paths with fixed starting points and unfixed ending points has been widely used to resolve many challenging problems in enumerative combinatorics. In this talk, we present a new formula that complements Okada and Stembridge's Pfaffian formula. The combinatorial interpretation of the new formula gives a reflection principle for nonintersecting paths. It implies that the enumeration of families of nonintersecting paths with unfixed ending points can be resolved by enumerating families of nonintersecting paths with fixed ending points instead. If time permits, we also explain how the enumeration of lozenge tilings of a large family of regions with free boundaries can be deduced from those without free boundaries, and present several applications of this result. Talk based on arXiv 2508.12135 slides

In this talk we introduce a new chip-firing model that makes close contact with the combinatorics of labeled and unlabeled rational-slope lattice paths. In generalizing the classic model, we extend chip-firing to graphs with rational edge weights while extending a number of important properties such as duality between superstable and recurrent configurations, enumerative results relating to the Catalan numbers and parking functions, and the existence of an associated critical group. Rational parking functions appear as the lattice-path analogues of those configurations on the complete graph that are supserstable with respect to this quantized rational chip-firing model.
Talk based on a joint work with Spencer Backman and Nick Loehr; see arXiv 2603.15451. slides

The fully inhomogeneous spin Hall-Littlewood symmetric rational functions are the partition functions of tuples of lattice paths consisting of north and east steps with certain boundary conditions in the $\mathfrak{sl}_2$ higher spin six vertex model. For a special choice of the parameters in the vertex weights, we obtain tuples of non-intersecting lattice paths and show that these are in bijection with rectangular alternating sign matrices. We present two new Littlewood identities for the fully inhomogeneous spin Hall-Littlewood symmetric rational functions, generalising various known identities for Schur polynomials, Hall-Littlewood P polynomials and the modified Robbins polynomials. This is joint work with Ilse Fischer from the University of Vienna: arXiv 2603.29836. slides

Progressive paths and rushed paths are two families of Dyck paths studied by Asinowski and Jelínek, which have the same enumerating sequence (OEIS entry A287709). I will present a bijection proving this fact. Rushed paths turn out to be in bijection with one-sided trees, introduced by Durhuus and Ünel, which have an asymptotic enumeration involving a stretched exponential. I will show some asymptotic results and present a random sampling procedure for rushed paths. I will conclude by presenting several other classes of related lattice paths and directed animals that may have similar asymptotic properties. Talk based on arXiv 2403.08120. slides

We address the Bousquet-Mélou-Mishna conjecture, which predicts that the counting generating function $Q(1,1;t)$ of a small-step quarter-plane lattice model is D-finite if and only if the group of the walk is finite. The finite-group case is complete, and the infinite-group result for the multivariate endpoint generating function $Q(x,y;t)$ has recently been resolved. However, the specialization to $Q(1,1;t)$ remains open in general. We organize the infinite-group models according to what is known and conjectured about its non-D-finiteness.
For $21$ of the $56$ infinite-group models — the five singular models, three with zero drift, and thirteen with polar interior drift — non-D-finiteness follows from asymptotic results of Bostan–Raschel–Salvy and probabilistic estimates of Denisov–Wachtel and Duraj. Nine further models have differentially algebraic endpoint generating functions via decoupling functions, though their D-finiteness remains open. For 21 additional models, we propose a new approach based on numerical estimates of singular exponents of the boundary series $Q(1,0;t)$ and $Q(0,1;t)$, and state conjectures toward completing the classification. We also discuss ongoing progress toward proving some of these conjectures. Talk based on a joint work with Marni Mishna: arXiv 2605.16688. slides

The enumeration of walks in the quarter plane confined in the first quadrant has attracted a lot of attention over the past twenty years. The generating functions associated to small steps models satisfy a functional equation in two catalytic variables. For such models, Bousquet-Mélou and Mishna, along Fayolle, Iasnogorodski and Malyshev defined a group called the group of the walk which turned out to be central in the classification of small steps models. In particular, its action on the catalytic variables yields a set of change of variables compatible with the structure of the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer, with no underlying group. In this talk, I will explain how one can endow the orbit with the action of a Galois group, which extends the group of the walk to models with large steps. The talk will then focus on the first major application of this Galoisian framework to the study of walks, that is a generic algebraicity strategy for walks with small backwards steps. This strategy is based Tuttes invariants method, introduced by Bernardi, Bousquet-Mélou and Raschel. Namely, the galois theoretic formulation allows for a complete characterization and contruction of rational invariants and decouplings, that are crucial for this strategy. In the end, this allows to prove the algebraicity of many models with large steps. Talk based (in part) on a joint work with Charlotte Hardouin: arXiv 2409.11084. slides

Tutte’s invariant method as introduced by Mireille Bousquet-Mélou, Olivier Bernardi and Kilian Raschel provides uniform algebraicity proofs for generating functions of restricted lattice walks. Whether it applies or not very much relies on the existence of rational functions that are called invariants and decoupling functions. I will present the ideas that underly Tutte's invariant method, explain how invariants can be constructed, and discuss a differential analogue that relies on invariants only. Talk based (in part) on a joint work with Charlotte Hardouin, and on earlier work: arXiv 2411.08584. slides

In a recent paper with Richard Ehrenborg and Margaret Readdy (arXiv 2511.04815), we expressed the toric $g$-vector entries of any simple polytope as a nonnegative integer linear combination of its gamma-vector entries. We showed that the toric $g$-vector of the associahedron is the ascent statistic of all 123-avoiding parking functions. An analogous result holds for the cyclohedron and 123-avoiding functions. We also proved that the toric $g$-vector of the permutahedron records the ascent statistics of parking trees representing 123-avoiding parking functions. Our approach extends to all chordal nestohedra. In this talk we will focus on the associahedron, where our proof relies on pairing the Dyck path representation of a parking function by Garsia and Haiman with the Dyck path representation of a 123-avoiding permutation by Krattenthaler. slides

We provide a complete combinatorial and asymptotic analysis of positive linear systems of equations in one catalytic variable that appear in several combinatorial problems, such as lattice-path counting or stack-sortable permutation counting. We show that the corresponding generating functions satisfy a positive polynomial system of equations (which is associated to a context-free grammar). Furthermore we prove a universal asymptotic behaviour. Talk based on joint work with Cyril Banderier: arXiv 2605.22241. slides

In the classical symbolic method, one is given a countable family $A$ of combinatorial objects, a size function $s$, and a weight function $h$ from $A$ to a ring $R$. The coefficients in the generating function corresponding to $A$, $s$, and $h$ are defined as the finite sums of $h$-weights of objects $a \in A$ with fixed size $s(a)=n$. In our extension of the symbolic method, we drop the finiteness condition, and we only require that the ring R is endowed with a complete norm and that every coefficient sum absolutely converges. We exemplify our extension on generalizations of classical Pólya's theorem about random walks in the d-dimensional grid. Talk based on a joint work with Richard Horský: arXiv 2511.00914. slides

The study of $q,t$-Catalan polynomials is a central topic in algebraic combinatorics, deeply connected to statistics on Dyck paths. In 2021, Pappe, Paul, and Schilling introduced two novel statistics, depth and ddinv, for classical Dyck paths, elegantly establishing the $q,t$-symmetry of the joint distributions of (area, depth) and (dinv, ddinv) via involutions on plane trees. In this talk, guided by the observation that depth is essentially a variant of Xin and Zhang's earlier bounce statistic, we extend this concept to the broader framework of $\vec{k}$-Dyck paths. We establish the $q,t$-symmetry of (area, depth) for these paths, which, as a byproduct, yields an alternative interpretation of the higher $q,t$-Catalan polynomials. Talk based on joint work with Yingrui Zhang: arXiv 2510.08196. slides

Alt $\nu$-Tamari lattices constitute a remarkable family of lattices associated with lattice paths that broadly generalize the Dyck and Tamari lattices. To systematically study the structural properties of this family, we introduce a combinatorial model that realizes the canonical join complex of alt $\nu$-Tamari lattices. Serving as a universal tool, this model allows us to prove vertex decomposability, establish an explicit shelling order, and reveal the underlying homology of the canonical join complex of alt $\nu$-Tamari lattices. Talk based on arXiv 2605.13770. slides

Consider a short uniform random walk in $\mathbb{R}^d$ that starts at the origin and consists of $m$ independent unit steps in random directions (this walk is also known in the literature as random flights). Take the distance to the origin (after $m$ such steps) and compute a sequence of its even moments. As was shown in 2015 by Borwein, Straub, and Vignat, in dimensions $d=2$ and $d=4$, this is an integer sequence. While for $d=2$, the $2n$th moment is equal to the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known.
The aim of this talk is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths with a prescribed number of peaks and words of a certain type. In addition, this bijection allows us to derive closed formulas for the number of lattice paths provided with certain statistics. Talk based on a work with Sergey Kirgizov and Michael Wallner: arXiv 2605.16715. slides

The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this work, we give a particular infinite family of presentations for which the cogrowth series can be determined as the constant term of an algebraic function, which shows that it is D-finite and, with more work, not algebraic. Our proof exploits the fact that for a particular choice of subgroup, the corresponding Schreier graph has finite tree width, and by considering paths in the cosets and the Schreier graph separately, we are able to construct a system of generating functions which count paths. We find the asymptotics of this system to conclude that the groups have D-finite but non-algebraic cogrowth series. Talk based on a joint work with Murray Elder and Andrew Rechnitzer: arXiv 2605.12793. slides

We present a novel bijection between stacked directed polyominoes and Motzkin paths with catastrophes. Further, we leverage this new bridge between these two worlds to obtain a better understanding of certain parameters of stacked directed animals. In particular, we obtain improved lower and upper bounds on the asymptotic width of stacked directed animals. This is joint work with Michael Wallner: arXiv 2406.16417. slides

Tree-child networks form an important subclass of phylogenetic networks, used to model reticulate evolutionary processes. The Pons-Batle conjecture asserts an identity between a class of words of length $2n+k$ and the set of tree-child networks with $n$ leaves and $k$ reticulation nodes. Although the conjecture has been proved by computational methods, a combinatorial explanation is still missing.
In this talk, we study this conjecture through lattice paths and generating functions. We associate the relevant objects with two distinct families of lattice paths, and translate the resulting path models into generating functions. Using differential operators for one family and a recursive algorithm based on the number of reticulation nodes for the other family, we obtain an alternative verification of the conjecture for all fixed $k \le 250$.
All these generating functions are algebraic and the asymptotic properties of the associated sequences can be analyzed using singularity analysis. In particular, we can analyze the limit laws of several natural parameters of the networks. In the cases we consider, the limiting distributions are Beta and Uniform distributions.
Talk based on joint work with Guan-Ru Yu (National Sun Yat-sen University) and Michael Wallner (TU Wien): arXiv 2605.07587. slides

We show that the global dimension of linear Nakayama algebras has the same distribution as the height of Dyck paths. To do so, we consider a slight modification of Ringel's resolution quiver and a mix of enumerative and bijective arguments. Talk based on a joint work with Viktória Klász, René Marczinzik, Anton Mellit, and Christian Stump: arXiv 2503.18415. slides

What happens when lattice paths are replaced by walks on a branching structure? Tree walks are a family of combinatorial objects whose enumeration is closely connected to moment calculations in random matrix theory. While the underlying definition is simple, the resulting counting problems are remarkably rich and many questions remain unresolved. In this talk, I will introduce tree walks, explain their connection to spectral distributions of random matrices, and survey recent results and open problems. The talk is based on arXiv 2405.08347 and arXiv 2508.08865. slides