Michael Wallner

Florian Schager, Michael Wallner

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.

Élie de Panafieu, Michael Wallner

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. Keywords. Random walks, analytic combinatorics, generating functions, limit laws, networking, encapsulation.

Michael Wallner

A chain is defined as a directed acyclic graph (DAG) with one source and one sink, where the children are ordered and the spanning tree computed using a depth-first search is a path. Such DAGs emerge in the context of tree compression and are therefore uniquely associated with a tree. The tree size of a DAG is defined as the size of the associated tree. For fixed out-degree $k \geq 2$, we compute the asymptotic expected decompressed tree size of a chain of size $n$ chosen uniformly at random, and we show that it contains a stretched exponential term of the form $e^{c \, \sqrt{n}}$. This result also has implications for the limit distribution of Brauer chains of fixed length.

Hexuan Liu, Michael Wallner, Guan-Ru Yu

Tree-child networks are an important class of phylogenetic network used to model reticulate evolutionary processes. These networks have attracted increasing attention from researchers with interests in both combinatorics and algorithms. A fundamental open problem posed by Pons and Batle asks whether the number $TC_{n,k}$ of bicombining tree-child networks with $n$ leaves and $k$ reticulation nodes equals the number of certain constrained words, now called Pons-Batle words. In this paper, we confirm the conjecture for tree-child networks with a bounded number of reticulation nodes. Our approach is combinatorial and analytic. We introduce families of Young tableaux with walls and holes and construct explicit bijections with Pons-Batle words, yielding a direct combinatorial explanation of the identities. These tableaux encode structural features of the underlying networks, including the placement of reticulation nodes. By projecting them to decorated Dyck paths, we obtain algebraic generating functions with differential operators encoding step weights, leading to explicit recurrence relations and closed-form formulas for $TC_{n,k}$. Beyond finite verification for moderate $k$, the framework reveals an underlying probabilistic structure. For $k=1$, natural structural parameters, such as the position and value of distinguished cells, converge, after rescaling, to $\mathrm{Beta}(2,1)$, $\mathrm{Beta}(1,2)$, and Uniform (i.e., $\mathrm{Beta}(1,1)$) distributions. These limit laws arise from a coalescence of singularities at the dominant square-root singularity, producing a non-analytic transition in the local expansion. Overall, our results provide both combinatorial insight and a unified analytic perspective on the asymptotic behavior of tree-child networks, showing how algebraic generating functions with interacting singularities systematically produce Beta limit laws.

Lukas Spiegelhofer, Michael Wallner

The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\leq t<2^k-1$. Then \[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leq 2^{k-1}.\] We prove that the Tu--Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$. Moreover, we prove that the Tu--Deng Conjecture implies a conjecture due to T.~W.~Cusick concerning the sum of digits of $n$ and $n+t$.