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Stephan Wagner

Publications and source records attributed to Stephan Wagner.

84 records · Page 5Linked to original sources
arXiv2015

Uniform spanning trees on Sierpinski graphs

We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution. The connection between uniform spanning trees and loop-erased random walk is then exploited to prove convergence of the latter to a continuous stochastic process. Some geometric properties of this limit process, such as the Hausdorff dimension, are investigated as well. The method is also applicable to other self-similar graphs with a sufficient degree of symmetry.

math.PR↗
arXiv2014

Compositions into Powers of $b$: Asymptotic Enumeration and Parameters

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated.

math.NT↗
arXiv2014

Variances and Covariances in the Central Limit Theorem for the Output of a Transducer

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-$w$ non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence.

math.CO↗
arXiv2014

Enumeration of the adjunctive hierarchy of hereditarily finite sets

Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. This raises the need for efficient representation of such sets, for example by numbers. In 2008, Kirby proposed an adjunctive hierarchy of hereditarily finite sets, based on the fact that they can also be seen as built up from the empty set by repeated adjunction, that is, by the addition of a new single element drawn from the already existing sets to an already existing set. Determining the cardinality $a_n$ of each level of this hierarchy, problem crucial in establishing whether the natural adjunctive hierarchy leads to an efficient encoding by numbers, was left open. In this paper we solve this problem. Our results can be generalized to hereditarily finite sets with atoms, or can be further refined by imposing restrictions on rank, on cardinality, or on the maximum level from where the new adjoined element can be drawn. We also show that $a_n$ satisfies the asymptotic formula $a_n = C^{2^n} + O(C^{2^{n-1}})$, for a constant $C \approx 1.3399$, which is a too fast asymptotic growth for practical purposes. We thus propose a very natural variant of the adjunctive hierarchy, whose asymptotic behavior we prove to be $Θ(2^n)$. To our knowledge, this is the first result of this kind.

cs.LO↗
arXiv2013

The number of fixed points of Wilf's partition involution

Wilf partitions are partitions of an integer $n$ in which all nonzero multiplicities are distinct. On his webpage, the late Herbert Wilf posed the problem to find "any interesting theorems" about the number f(n) of those partitions. Recently, Fill, Janson and Ward (and independently Kane and Rhoades) determined an asymptotic formula for $\log f(n)$. Since the original motivation for studying Wilf partitions was the fact that the operation that interchanges part sizes and multiplicities is an involution on the set of Wilf partitions, they mentioned as an open problem to determine a similar asymptotic formula for the number of fixed points of this involution, which we denote by F(n). In this short note, we show that the method of Fill, Janson and Ward also applies to F(n). Specifically, we obtain the asymptotic formula $\log F(n) \sim \frac12 \log f(n)$.

math.CO↗
arXiv2013

Spectral moments of trees with given degree sequence

Let $λ_1,\dots,λ_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=λ_1^k+\dots+λ_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \dots highest degrees to the neighbors of the root, and so on) for any $k\geq 0$. Several corollaries follow, in particular a conjecture of Ilić and Stevanović on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Marković and Glišić on the Estrada index of such trees, which is defined as $\EE(G)=e^{λ_1}+\dots+e^{λ_n}$.

math.CO↗
arXiv2012

Asymptotics of generalised trinomial coefficients

It is shown how to obtain an asymptotic expansion of the generalised central trinomial coefficient $[x^n](x^2 + bx + c)^n$ by means of singularity analysis, thus proving a conjecture of Zhi-Wei Sun.

math.NT↗
arXiv2011

Labeled trees, maps, and an algebraic identity

We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher label. This solves a problem posed by Shin and Zeng in a recent article. We also provide a generalization of this identity that translates to a formula for the number of rooted spanning forests with given indegree sequence.

math.CO↗
arXiv2011

Free Lamplighter Groups and a Question of Atiyah

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.

math.GR↗
arXiv2010

The number of maximum matchings in a tree

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupień on the number of maximal matchings (maximal with respect to set inclusion).

math.CO↗
arXiv2006

Calculating the correlation coefficients of graph-theoretical indices

Using a generating function approach, the correlation coefficients of four different graph-theoretical indices, namely the number of independent vertex subsets, the number of matchings, the number of subtrees and the Wiener index, are asymptotically determined for random rooted ordered trees.

math.CO↗