arXiv subjects

Stephan Wagner

Publications and source records attributed to Stephan Wagner.

At least 37 records · Page 2Linked to original sources

arXiv2024

A bijection for the evolution of $B$-trees

A $B$-tree is a type of search tree where every node (except possibly for the root) contains between $m$ and $2m$ keys for some positive integer $m$, and all leaves have the same distance to the root. We study sequences of $B$-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with $B$-trees.

math.CO↗

arXiv2023

Local limits of descent-biased permutations and trees

We study two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation $σ$ is a pair of consecutive elements $σ(i), σ(i+1)$ such that $σ(i) > σ(i+1)$. Likewise, a descent in a rooted tree with labelled vertices is a pair of a parent vertex and a child such that the label of the parent is greater than the label of the child. For some nonnegative real number $q$, we consider the probability measures on permutations and on rooted labelled trees of a given size where each permutation or tree is chosen with a probability proportional to $q^{\text{number of descents}}$. In particular, we determine the asymptotic distribution of the first elements of permutations under this model. Different phases can be observed based on how $q$ depends on the number of elements $n$ in our permutations. The results on permutations then allow us to characterize the local limit of descent-biased rooted labelled trees.

math.PR↗

arXiv2023

Bounding mean orders of sub-$k$-trees of $k$-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.

math.CO↗

arXiv2023

The distribution of the number of automorphisms of random trees

We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.

math.PR↗

arXiv2023

Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean

For a complex number $α$, we consider the sum of the $α$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(α)$ have been determined for $\Reα\neq 0$, revealing a transition between a complex normal limiting distribution for $\Reα< 0$ and a non-normal limiting distribution for $\Reα> 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Reα= 0$. The same results are also established in the case of the so-called shape functional $X_n'(0)$, which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. Additionally, we prove convergence of all moments in the case $\Reα< 0$, where this result was previously missing, and establish new results about the asymptotic mean for real $α< 1/2$. A novel feature for $\Reα=0$ is that we find joint convergence for several $α$ to independent limits, in contrast to the cases $\Reα\neq0$, where the limit is known to be a continuous function of $α$. Another difference from the case $\Reα\neq0$ is that there is a logarithmic factor in the asymptotic variance when $\Reα=0$; this holds also for the shape functional. The proofs are largely based on singularity analysis of generating functions.

math.PR↗

arXiv2023

The Uncover Process for Random Labeled Trees

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this work: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition.

math.PR↗

arXiv2022

On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue $α$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $μ_α n$ and $σ^2_α n$ respectively. It is also shown that $μ_α$ and $σ^2_α$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $μ_0$ and $σ^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

math.CO↗

arXiv2022

Refined enumeration of $k$-plane trees and $k$-noncrossing trees

A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to $(2k+1)$-ary trees.

math.CO↗

arXiv2022

Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-steps at a height of $i$ modulo $k$. Given a tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration formula for the number of $k_t$-Dyck paths of length $(k+1)n$ with $a_i$ down-steps at a height of $i$ modulo $k$, $1 \leq i \leq k$. The proofs given are done via bijective means or with generating functions.

math.CO↗

arXiv2022

The birth of the strong components

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + μn^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as $μ$ goes from $-\infty$ to $\infty$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of $μ$ and provide more properties of the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter $p$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2-cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

math.CO↗

arXiv2022

A polynomial associated with rooted trees and specific posets

We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other is new. For the third specialisation, we construct non-isomorphic trees with the same associated polynomial. We finally find that our polynomial can be generalised in a natural way to a family of posets that we call $\mathcal{V}$-posets. These posets are obtained recursively by either disjoint unions or adding a greatest/least element to existing $\mathcal{V}$-posets.

math.CO↗

arXiv2021

Trees with minimum number of infima closed sets

Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also provided.

math.CO↗

arXiv2021

Convex characters, algorithms and matchings

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. In \cite{KelkS17} it was shown how to efficiently count, list and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterised algorithms can be used to speed up exponential-time algorithms for the \emph{maximum agreement forest problem} in phylogenetics. Second, we re-visit the quantity $g_2(T)$, defined as the number of convex characters on $T$ in which each state appears on at least 2 taxa. We use this to give an algorithm with running time $O( ϕ^{n} \cdot \text{poly}(n) )$, where $ϕ\approx 1.6181$ is the golden ratio and $n$ is the number of taxa in the input trees, for computation of \emph{maximum parsimony distance on two state characters}. By further restricting the characters counted by $g_2(T)$ we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to $O( 1.5895^{n} \cdot \text{poly}(n) )$, and obtain a number of new results in themselves relevant to enumeration of matchings on at-most binary trees.

math.CO↗

arXiv2021

Broadcasting induced colourings of random recursive trees and preferential attachment trees

In this work we consider random two-colourings of random linear preferential attachment trees, which includes random recursive trees, random plane-oriented recursive trees, random binary search trees, and a class of random $d$-ary trees. The random colouring is defined by assigning the root of the tree the colour red or blue with equal probability, and all other vertices are assigned the colour of their parent with probability $p$ and the other colour otherwise. These colourings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices of each colour, the number of monochromatic subtrees of each colour, as well as the number of leaves and fringe subtrees with two-colourings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root. The description of the limiting distributions extends previous work on bond percolation in random preferential attachment trees.

math.PR↗

arXiv2021

On the maximum mean subtree order of trees

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we do not completely resolve this conjecture, we find some evidence in its favor by proving different features of trees that attain the maximum. For example, we show that the diameter of a tree of order $n$ with maximum mean subtree order must be very close to $n$. Moreover, we show that the maximum mean subtree order is equal to $n - 2\log_2 n + O(1)$. For the local mean subtree order, which is the average order of all subtrees containing a fixed vertex, we can be even more precise: we show that its maximum is always attained by a broom and that it is equal to $n - \log_2 n + O(1)$.

math.CO↗

arXiv2021

Distinct Fringe Subtrees in Random Trees

A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in two types of random trees: simply generated trees and families of increasing trees (recursive trees, $d$-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what `distinct' means) in random trees with $n$ vertices is $n/\sqrt{\log n}$ for simply generated trees and $n/\log n$ for increasing trees.

math.CO↗

arXiv2021

The number of distinct adjacent pairs in geometrically distributed words

A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($Γ_1, Γ_2, \dots, Γ_n$) where $\mathbb{P}(Γ_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$.

math.CO↗

arXiv2020

Irrationality of growth constants associated with polynomial recursions

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A α^{d^n}$, but little can be said about the constant $α$. In this paper, we show that $α$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A α^n + B + O(α^{-εn})$, where $A,B$ are algebraic and $α> 1$, and the sequence contains infinitely many integers, then $α$ is irrational or an integer.

math.NT↗