Michael Wallner

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n−1}$. These give relations between different statistics, such as the number of crossings of the x-axis in classes of Dyck bridges or the distribution of peaks in classes of Dyck paths, and furthermore relate them with k- and g-compositions. These allow us to find and prove congruence results for Dyck paths and parity results for compositions. Our investigation uncovers unexpected connections to mock theta functions, Hardinian arrays, little Schröder paths, Fibonacci numbers, and irreducible pairs of compositions, offering new insights into the structures of paths, partitions and compositions.

We give bijective results between several variants of lattice paths and integer compositions all enumerated by the seemingly innocuous formula $4^{n-1}$. These associations lead us to make more connections between the two constructs such as congruence results.

Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for {\it bicombining tree-child networks} in which every reticulation node has exactly two parents. We extend these studies to {\it $d$-combining tree-child networks} where every reticulation node has now $d\geq 2$ parents, and we study one-component as well as general tree-child networks. Moreover, we also give results on the distributional behavior of shape parameters (e.g., number of reticulation nodes, Sackin index) of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves. We show phase transitions depending on $d$, leading to normal, Bessel, Poisson, and degenerate distributions. Some of our results are new even in the bicombining case.

Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to $d$-combining tree-child networks where every reticulation node has now $d\geq 2$ parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves.

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.

We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-Mélou 2016]. We solve in detail a new case, where all 8 nearest neighbour steps are allowed. As in the two cases solved in [Bousquet-Mélou 2016], the associated generating function is proved to differ from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-Mélou 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.

We also explain why we expect the observed algebraicity phenomenon to persist for 4 more models, for which the quadrant problem is solvable using the reflection principle.

This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference to classical walks is that its nondeterministic steps consist of sets of steps from a predefined set such that all possible extensions are explored in parallel. We discuss in detail the most natural nondeterministic step sets (Dyck and Motzkin step sets), and show that several nondeterministic classes of lattice paths, such as nondeterministic bridges, excursions, and meanders are algebraic.

The key concept is the generalization of the ending point of a walk to its reachable points, i.e., a set of ending points. We extend our results to general step sets: We show that nondeterministic bridges and several subclasses of nondeterministic meanders are always algebraic. We conjecture the same is true for nondeterministic excursions, and we present python and Maple packages to support our 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.

Un tunnel est une portion d'un chemin où un protocole de communication est encapsulé dans un autre. Les tunnels sont de plus en plus présents dans les réseaux. Ils sont utilisés pour la sécurité, l'interopérabilité, les réseaux virtuels, etc. Un chemin contenant un tunnel doit ^etre faisable, c'est-à-dire qu'il doit respecter l'ordre des encapsulations et des désencapsulations, parfois imbriquées. Les algorithmes de calcul de chemins avec prise en compte de tunnels ne sont pas les m^emes que dans un réseau classique. Afin de tester ces algorithmes, il serait utile de disposer d'un générateur de topologies aléatoires avec tunnels. Ce générateur devra produire des topologies ayant certaines caractéristiques (par exemple, que la probabilité qu'un chemin soit faisable soit un paramètre p quelconque). Il se trouve que ce problème est lié aux chemins de Dyck et de Motzkin en combinatoire. En effet, une encapsulation peut ^etre vue comme un pas montant et une désencapsulation peut ^etre vue comme un pas descendant. La question qui se pose est donc de savoir quelle est la probabilité qu'un chemin de Dyck ou de Motzkin ait certaines propriétés. Dans ce travail, en utilisant des outils de combinatoire analytique, nous donnons la probabilité qu'un chemin soit faisable en fonction de la distribution des encapsulations et des désencapsulations sur les sommets.

This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel. We introduce our new model on Dyck steps with the nondeterministic step set $\{ \{-1\}, \{1\}, \{-1,1\}\}$ and Motzkin steps with the nondeterministic step set $\{\{-1\}, \{0\}, \{1\}, \{-1,0\}, \{-1,1\}, \{0,1\}, \{-1,0,1\}\}$.

For general lists of step sets and a given length, we express the generating function of nondeterministic walks where at least one of the walks explored in parallel is a bridge (ends at the origin). In the particular cases of Dyck and Motzkin steps, we also compute the asymptotic probability that at least one of those parallel walks is a meander (stays nonnegative) or an excursion (stays nonnegative and ends at the origin).

This research is motivated by the study of networks involving encapsulations and decapsulations of protocols. Our results are obtained using generating functions and analytic combinatorics.

Pólya urns are urns where at each unit of time a ball is drawn and replaced with some other balls according to its colour. We introduce a more general model: the replacement rule depends on the colour of the drawn ball and the value of the time (mod p). We extend the work of Flajolet et al. on Pólya urns: the generating function encoding the evolution of the urn is studied by methods of analytic combinatorics.

We show that the initial differential equations lead to ordinary linear differential equations which are related to hypergeometric functions (giving the exact state of the urns at time n). When the time goes to infinity, we prove that these periodic Pólya urns have asymptotic fluctuations which are described by a product of generalized gamma distributions. With the additional help of what we call the density method (a method which offers access to enumeration and random generation of poset structures), we prove that the law of the south-east corner of a triangular Young tableau follows asymptotically a product of generalized gamma distributions. This allows us to tackle some questions related to the continuous limit of large random Young tableaux and links with random surfaces.

Pólya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time mod p. We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial non-linear partial differential equation indeed leads to linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time n). When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions.

En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.

We present a general theorem on the structure of bivariate generating functions which gives sufficient conditions such that the limiting probability distribution is a half-normal distribution. If $X$ is a normally distributed random variable with zero mean, then $|X|$ obeys a half-normal distribution. In the second part, we apply our result to prove three natural appearances in the domain of lattice paths: the number of returns to zero, the height, and the sign changes are under zero drift distributed according to a half-normal distribution. This extends known results to a general step set. Finally, our result also gives a new proof of Banach's matchbox problem.

We present an extension of a theorem by Michael Drmota and Michele Soria [Images and Preimages in Random Mappings, 1997] that can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytical and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of Motzkin paths.

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size. Finally, we extend all results to other P\'{o}lya structure.

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is similar to having the possibility of jumping from any altitude to zero. These objects have the interesting feature: They do not have the same intuitive probabilistic behaviour as classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory). This article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and we make a connection with a continued fraction expansion. Furthermore, we prove several formulae for related combinatorial structures conjectured in the On-Line Encyclopedia of Integer Sequences. We use the kernel method and other tools from analytic combinatorics in order to provide the enumeration and limit laws of these lattice paths with catastrophes for all finite sets of jumps. We conclude by providing an algorithm which generates such lattice paths uniformly at random.

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not have the same intuitive probabilistic behavior like classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory), and this article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and a link with a continuous fraction expansion, and prove several formulae for related combinatorial structures conjectured in the On-line Encyclopedia of Integer Sequences. Thanks to the kernel method and via analytic combinatorics, we derive the enumeration and limit laws of these "lattice paths with catastrophes", for any finite set of jumps.

We analyze some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope $2/5$. This answers Knuth's problem #4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities. A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovsek. We show how to obtain similar results for other slopes than $2/5$. An interesting case being e.g. Dyck paths below the slope $2/3$, which corresponds to the so called Duchon's club model. We also show how to use our techniques for lattice paths below an irrational slope and illustrate this with Dyck paths below $y=x/\sqrt{2}$.

We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem #4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014.Our approach relies on the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths. A key ingredient in the proof is the generalization of an old trick of Knuth himself (for enumerating permutations sortable by a stack),promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic,and they offer some new combinatorial identities, which can be also tackled in the A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar results for other slopes than 2/5, an interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model.

In this abstract we analyze directed lattice paths, when a boundary reflecting or absorbing condition is added to the classical models. The lattice paths are characterized by two time-independent sets of rules (also called steps) which have a privileged direction of increase. Depending on the spatial coordinate, one of the two sets of rules applies, namely one for altitude 0 and one for altitude bigger than 0. The abscissa y = 0 thus acts as a border which either absorbs or reflects steps. The absorption model extends the model analyzed by Banderier and Flajolet [Basic analytic combinatorics of directed lattice paths, 2002], while the reflection model leads to a more complicated situation. More details are given in [Some reflections on lattice paths, 2014].

This article analyzes directed lattice paths, when a boundary reflecting or absorbing condition is added to the classical models. The lattice paths are characterized by two time-independent sets of rules (also called steps) which have a privileged direction of increase and are therefore essentially one-dimensional objects. Depending on the spatial coordinate, one of the two sets of rules applies, namely one for altitude 0 and one for altitude bigger than 0. The abscissa y=0 thus acts as a border which either absorbs or reflects steps. The absorption model corresponds to the model analyzed by Banderier and Flajolet ("Analytic combinatorics of directed lattice paths"), while the reflecting model leads to a more complicated situation. We show how the generating functions are then modified: the kernel method strikes again but here it unfortunately does not give a nice product formula. This makes the analysis more challenging, and, in the case of Lukasiewicz walks, we give the asymptotics for the number of excursions, arches and meanders. Limit laws for the number of returns to 0 of excursions are given. We also compute the limit laws of the final altitude of meanders. The full analytic situation is more complicated than the Banderier-Flajolet model (partly because new "critical compositions" appear, forcing us to introduce new key quantities, like the drift at 0), and we quantify to what extent the global drift, and the drift at 0 play a role in the "universal" behavior of such walks.