scholarly journals The height of watermelons with wall - Extended Abstract

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Thomas Feierl
Keyword(s):  
Average Height ◽  
Height Distribution ◽  
Famous Result ◽  
Plane Trees ◽  
International Audience ◽  
De Bruijn

International audience We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.

A Topographic Drag Closure Built on an Analytical Base Flux

2005 ◽  
Vol 62 (7) ◽  
pp. 2302-2315 ◽  
Author(s):  
Stephen T. Garner
Keyword(s):  
Dimensional Analysis ◽  
Wave Train ◽  
Average Height ◽  
Height Distribution ◽  
Ocean Tides ◽  
Linear Solution ◽  
Flow Over Topography ◽  
Linear Drag ◽  
The Vertical Distribution ◽  
State Assumption

Abstract Topographic drag schemes depend on grid-scale representations of the average height, width, and orientation of the subgrid topography. Until now, these representations have been based on a combination of statistics and dimensional analysis. However, under certain physical assumptions, linear analysis provides the exact amplitude and orientation of the drag for arbitrary topography. The author proposes a computationally practical closure based on this analysis. Also proposed is a nonlinear correction for nonpropagating base flux. This is patterned after existing schemes but is better constrained to match the linear solution because it assumes a correlation between mountain height and width. When the correction is interpreted as a formula for the transition to saturation in the wave train, it also provides a way of estimating the vertical distribution of the momentum forcing. The explicit subgrid height distribution causes a natural broadening of the layers experiencing the forcing. Linear drag due to simple oscillating flow over topography, which is relevant to ocean tides, has almost the same form as for the stationary atmospheric problem. However, dimensional analysis suggests that the nonpropagating drag in this situation is mostly due to topographic length scales that are small enough to keep the steady-state assumption satisfied.

scholarly journals Concentration Properties of Extremal Parameters in Random Discrete Structures

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Maximum Degree ◽  
Random Trees ◽  
Height Distribution ◽  
Exponential Tail ◽  
Scale Free ◽  
Main Emphasis ◽  
Discrete Structures ◽  
International Audience ◽  
Concentration Properties

International audience The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.

scholarly journals Numerical Studies of the Asymptotic Height Distribution in Binary Search Trees

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Charles Knessl
Keyword(s):  
Integral Equation ◽  
Binary Search ◽  
Linear Integral Equation ◽  
Height Distribution ◽  
Binary Search Trees ◽  
Search Trees ◽  
Asymptotic Results ◽  
Numerical Studies ◽  
International Audience ◽  
Linear Integral

International audience We study numerically a non-linear integral equation that arises in the study of binary search trees. If the tree is constructed from n elements, this integral equation describes the asymptotic (as n→∞) distribution of the height of the tree. This supplements some asymptotic results we recently obtained for the tails of the distribution. The asymptotic height distribution is shown to be unimodal with highly asymmetric tails.

scholarly journals The number of Euler tours of a random $d$-in/$d$-out graph

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Páidí Creed ◽  
Mary Cryan
Keyword(s):  
Asymptotic Distribution ◽  
Directed Graph ◽  
Polynomial Time ◽  
Simple Approach ◽  
Uniform Sampling ◽  
Concentration Result ◽  
International Audience ◽  
De Bruijn ◽  
Euler Tours

International audience In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.

scholarly journals The average height of planted plane trees with M leaves

1983 ◽  
Vol 34 (2) ◽  
pp. 191-208 ◽  
Author(s):  
R Kemp
Keyword(s):  
Average Height ◽  
Plane Trees

scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

scholarly journals Bootstrapping and double-exponential limit laws

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Maximum Degree ◽  
Exponential Rate ◽  
Limit Laws ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
Longest Run ◽  
Plane Trees ◽  
International Audience ◽  
Motzkin Paths

Combinatorics International audience We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.

scholarly journals Properties of the extremal infinite smooth words

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Srečko Brlek ◽  
Guy Melançon ◽  
Geneviève Paquin
Keyword(s):  
De Bruijn Graph ◽  
Lexicographical Order ◽  
Naive Algorithm ◽  
International Audience ◽  
De Bruijn

International audience Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.

scholarly journals The freeness of ideal subarrangements of Weyl arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Mohamed Barakat ◽  
Michael Cuntz ◽  
Torsten Hoge ◽  
Hiroaki Terao
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Height Distribution ◽  
Free Arrangement ◽  
Positive Roots ◽  
International Audience

International audience A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.

scholarly journals Local average height distribution of fluctuating interfaces

2017 ◽  
Vol 95 (1) ◽  
Author(s):  
Naftali R. Smith ◽  
Baruch Meerson ◽  
Pavel V. Sasorov
Keyword(s):  
Average Height ◽  
Height Distribution ◽  
Fluctuating Interfaces ◽  
Local Average
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