scholarly journals On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

scholarly journals Non-ambiguous trees: new results and generalization

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jean-Christophe Aval ◽  
Adrien Boussicault ◽  
Bérénice Delcroix-Oger ◽  
Florent Hivert ◽  
Patxi Laborde-Zubieta
Keyword(s):  
Binary Trees ◽  
Higher Dimension ◽  
International Audience ◽  
Definition Of ◽  
New Formula ◽  
Ential Equation

International audience We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

scholarly journals Introduction

2018 ◽  
Vol 27 (4) ◽  
pp. 441-441
Author(s):  
PAUL BALISTER ◽  
BÉLA BOLLOBÁS ◽  
IMRE LEADER ◽  
ROB MORRIS ◽  
OLIVER RIORDAN
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

scholarly journals The Explicit Laplace Transform for the Wishart Process

2014 ◽  
Vol 51 (3) ◽  
pp. 640-656 ◽  
Author(s):  
Alessandro Gnoatto ◽  
Martino Grasselli
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Time Integral ◽  
Variation Of Constants ◽  
The Matrix ◽  
Wishart Process ◽  
New Formula ◽  
Original Approach

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

scholarly journals The Confidence Density for Correlation

Sankhya A ◽  
2021 ◽  
Author(s):  
Gunnar Taraldsen
Keyword(s):  
Explicit Formula ◽  
Posterior Distribution ◽  
Hypergeometric Functions ◽  
Real Data ◽  
Numerical Calculations ◽  
Posterior Density ◽  
Fiducial Inference ◽  
Objective Prior ◽  
Confidence Distribution ◽  
New Formula

AbstractInference for correlation is central in statistics. From a Bayesian viewpoint, the final most complete outcome of inference for the correlation is the posterior distribution. An explicit formula for the posterior density for the correlation for the binormal is derived. This posterior is an optimal confidence distribution and corresponds to a standard objective prior. It coincides with the fiducial introduced by R.A. Fisher in 1930 in his first paper on fiducial inference. C.R. Rao derived an explicit elegant formula for this fiducial density, but the new formula using hypergeometric functions is better suited for numerical calculations. Several examples on real data are presented for illustration. A brief review of the connections between confidence distributions and Bayesian and fiducial inference is given in an Appendix.

scholarly journals The b-chromatic number of power graphs

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Brice Effantin ◽  
Hamamache Kheddouci
Keyword(s):  
Chromatic Number ◽  
Binary Trees ◽  
Proper Coloring ◽  
Power Cycles ◽  
International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

scholarly journals Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Christophe Aval ◽  
Philippe Duchon
Keyword(s):  
Alternating Sign Matrices ◽  
International Audience ◽  
Enumeration Formula ◽  
Half Turn

International audience The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestrited ASm's and the number of half-turn symmetric ASM's. L'objet de ce travail est d'énumérer les matrices à signes alternants (ASM) quasi-invariantes par rotation d'un quart-de-tour. La formule d'énumération, conjecturée par Duchon, fait apparaître trois facteurs, comprenant le nombre d'ASM quelconques et le nombre d'ASM invariantes par demi-tour.

scholarly journals A bijection for nonorientable general maps

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jérémie Bettinelli
Keyword(s):  
Simple Form ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Enumeration Formula

International audience We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.

scholarly journals Explicit and asymptotic formulae for Vasyunin-cotangent sums

2017 ◽  
Vol 102 (116) ◽  
pp. 155-174 ◽  
Author(s):  
Mouloud Goubi ◽  
Abdelmejid Bayad ◽  
Mohand Hernane
Keyword(s):  
Asymptotic Formula ◽  
Explicit Formula ◽  
Continued Fraction ◽  
Reciprocity Law ◽  
Asymptotic Formulae

For coprime numbers p and q, we consider the Vasyunin-cotangent sum V(q, p)= ?p?1 k=1 {kq/p} cot (?k/p). First, we prove explicit formula for the symmetric sum V(p,q)+ V(q,p) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for V(p,q). Finally, by use of continued fraction theory, we give a formula for V(p,q) in terms of continued fraction of p/q.

scholarly journals Tiling the Line with Triples

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Aaron Meyerowitz
Keyword(s):  
Greedy Algorithm ◽  
Direct Proof ◽  
Open Problems ◽  
One Dimensional ◽  
Proof By Contradiction ◽  
International Audience ◽  
The Subject ◽  
The One ◽  
Finite Digraph ◽  
Elegant Proof

International audience It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.

Sign in / Sign up

Export Citation Format

Share Document