scholarly journals Dyck paths, Motzkin paths and traffic jams

2004 ◽  
Vol 2004 (10) ◽  
pp. P10007 ◽  
Author(s):  
R A Blythe ◽  
W Janke ◽  
D A Johnston ◽  
R Kenna
Keyword(s):  
Traffic Jams ◽  
Dyck Paths ◽  
Motzkin Paths

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals A Short Approach to Catalan Numbers Modulo $2^r$

10.37236/664 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guoce Xin ◽  
Jing-Feng Xu
Keyword(s):  
Catalan Numbers ◽  
Binary Expansion ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Recent Classification ◽  
Motzkin Paths

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.

scholarly journals Deutsch paths and their enumeration

2021 ◽  
Vol 4 (1) ◽  
pp. 12-18
Author(s):  
Helmut Prodinger ◽  
Keyword(s):  
Arbitrary Length ◽  
Dyck Paths ◽  
Motzkin Paths

A variation of Dyck paths allows for down-steps of arbitrary length, not just one. Credits for this invention are given to Emeric Deutsch. Surprisingly, the enumeration of them is somewhat akin to the analysis of Motzkin-paths; the last section contains a bijection.

scholarly journals Minimal and maximal plateau lengths in Motzkin paths

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Gumbel Distribution ◽  
Minimal Length ◽  
Analytic Method ◽  
Horizontal Segment ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Substitution Process ◽  
International Audience ◽  
Motzkin Paths

International audience The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters <i> maximal plateau length, horizontal segment </i>and <i>maximal horizontal segment </i>as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

scholarly journals Path Counting and Random Matrix Theory

10.37236/1736 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Ioana Dumitriu ◽  
Etienne Rassart
Keyword(s):  
Random Walks ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Open Problem ◽  
Matrix Theory ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Motzkin Paths ◽  
Path Counting

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.

scholarly journals Counting peaks and valleys in $k$-colored Motzkin paths

10.37236/1913 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
A. Sapounakis ◽  
P. Tsikouras
Keyword(s):  
Fixed Number ◽  
Dyck Paths ◽  
Special Cases ◽  
Left And Right ◽  
High Level ◽  
Motzkin Paths

This paper deals with the enumeration of $k$-colored Motzkin paths with a fixed number of (left and right) peaks and valleys. Further enumeration results are obtained when peaks and valleys are counted at low and high level. Many well-known results for Dyck paths are obtained as special cases.

scholarly journals Avoiding patterns in irreducible permutations

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Jean-Luc Baril
Keyword(s):  
Fixed Point ◽  
Pattern Avoidance ◽  
Dyck Paths ◽  
Classical Pattern ◽  
International Audience ◽  
Motzkin Paths

International audience We explore the classical pattern avoidance question in the case of irreducible permutations, <i>i.e.</i>, those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.

Analisis Kandungan Mikroba, Formalin, dan Timbal (Pb) pada Tahu Sumedang yang Dijual Di Daerah Macet Cicurug, Ciawi, dan Cisarua Jawa Barat

2020 ◽  
Vol 6 (1) ◽  
pp. 087
Author(s):  
Rosy Hutami ◽  
M Fakih Kurniawan ◽  
Henna Khoerunnisa
Keyword(s):  
Content Analysis ◽  
Microbial Contamination ◽  
Atomic Absorption Spectrophotometry ◽  
Plate Count ◽  
Traffic Jam ◽  
Absorption Spectrophotometry ◽  
Traffic Jams ◽  
Total Plate Count ◽  
Microbial Analysis ◽  
Aas Method

Sumedang tofu is one of favorite foods for Indonesian society. But many sellers or producers are not aware to the food safety of sumedang tofu. The aims of this study were to analyze the microbial, formalin, and lead (Pb) contents in ready-to-eat sumedang tofu which were sold in traffic jams area in Cicurug, Ciawi, and Cisarua. The analysis were carried out by Total Plate Count (TPC) testing for microbial analysis, potassium permanganate reaction (KMnO4) testing for formaldehyde analysis, and atomic absorption spectrophotometry (AAS) method for lead content analysis in the samples. The results obtained for the microbial analysis were sumedang tofu that were sold in the traffic jam areas of Cicurug, Ciawi, and Cisarua contained contaminant above the treshold (1.4 x 105 colonies / gram to 2.2 x 105 colonies / gram of microbes). All of the samples of sumedang tofu were positive containing formaldehyde. Otherwise, there were no lead (Pb) content in all samples regarding to AAS analysis. This study concluded that the ready-to-eat sumedang tofu those were sold in traffic jam area in the Cicurug, Ciawi, and Cisarua were not suitable for consumption because it contained exceed microbial contamination and formalin which are harmful for human health.Keywords : formalin, microbes, sumedang tofu, lead, traffic jam

Complex Solution of Ecological and Economic Problems of Traffic Jams

2019 ◽  
pp. 6-15
Author(s):  
Oleksandr M. Matsenko ◽  
Yaroslav S. Kovalev ◽  
Olena M. Tkachenko ◽  
Yaroslava V. Chorna
Keyword(s):  
Traffic Control ◽  
Motor Vehicle ◽  
Motor Vehicles ◽  
Water Runoff ◽  
Negative Consequences ◽  
Traffic Lights ◽  
Economic Problems ◽  
Traffic Jams ◽  
Economic Sphere ◽  
Flows Over Time

The article explores the congestion level in traffic of motor vehicles and its negative environmental and economic consequences in case of Kiev. The amount of pollution from traffic jams in Kiev and the number of vehicles which got into them in 2009-2018 is analyzed. The loss of earnings on the side of automobile owners from their standby are calculated with corresponding quantitative expressions found and described. For the course of the research, the methods of system-structural and comparative analysis were used for analyzing the environmental and economic problems of modern automobile systems; methods of formal logical analysis were used for substantiating the innovative infrastructure of transport routes. Separately economic and statistical methods were used in the study for trends development, structure analysis, and estimation of the influence of road congestion on the environmental and economic sphere. Pearson test has indicated a close relationship between the number of cars in Kiev and the number of values from traffic jams in environmental and economic sphere. Solutions to this problem are offered in forms of automated traffic control systems, improvisation of organizational and technical methods for the distribution of traffic flows over time, namely reverse traffic, road junctions, smart traffic lights, road extension, and the transition to alternative modes of transport. In all countries of the world there are new research methods that affect pollutants from motor vehicles. It is proved that they are forced by the recipients. In addition, landscaping can improve landscape design, reduce greenhouse gas emissions, surface water runoff and noise pollution. In this regard the policy implication of the research are aimed to eliminate the negative consequences from the use of vehicles during traffic jams, and the necessary number of trees for planting in Kiev is calculated. Key words: motor transport, congestion, traffic jam, motor vehicle, greening, compensation effect, lost profits, losses.

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