scholarly journals Counting Fixed-Height Tatami Tilings

10.37236/215 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Frank Ruskey ◽  
Jennifer Woodcock
Keyword(s):  
Generating Functions ◽  
Binomial Coefficients ◽  
Explicit Solutions ◽  
Four Corners ◽  
Combinatorial Decomposition

A tatami tiling is an arrangement of $1 \times 2$ dominoes (or mats) in a rectangle with $m$ rows and $n$ columns, subject to the constraint that no four corners meet at a point. For fixed $m$ we present and use Dean Hickerson's combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when $n \ge m$. Using this decomposition we find the ordinary generating functions of both unrestricted and inequivalent tatami tilings that fit in a rectangle with $m$ rows and $n$ columns, for fixed $m$ and $n \ge m$. This allows us to verify a modified version of a conjecture of Knuth. Finally, we give explicit solutions for the count of tatami tilings, in the form of sums of binomial coefficients.

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials

2020 ◽  
pp. 42-42
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Recurrence Formula ◽  
Binomial Coefficients ◽  
Combinatorial Identity ◽  
Lucas Numbers ◽  
Fundamental Properties ◽  
Special Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers are given.

XIX.—Asymptotic enumeration of connected graphs

1970 ◽  
Vol 68 (4) ◽  
pp. 298-308
Author(s):  
E. M. Wright
Keyword(s):  
Asymptotic Expansion ◽  
Generating Functions ◽  
Binomial Coefficients ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Connected Graphs

SynopsisThe number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

Multisection of series

2016 ◽  
Vol 100 (549) ◽  
pp. 460-470
Author(s):  
Raymond A. Beauregard ◽  
Vladimir A. Dobrushkin
Keyword(s):  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Binomial Coefficients ◽  
Discrete Mathematics ◽  
Mathematical Tool ◽  
New Birth ◽  
Leonhard Euler ◽  
Finite Sums ◽  
Symbolic Methods ◽  
Infinite Sums

In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients.While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.

From golden-ratio equalities to Fibonacci and Lucas identities

2013 ◽  
Vol 97 (539) ◽  
pp. 234-241
Author(s):  
Martin Griffiths
Keyword(s):  
High School Students ◽  
Generating Functions ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
School Students ◽  
Simple Method ◽  
Matrix Methods ◽  
Binet’S Formula

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

The Process of Globalization and Its Impact on Media

2016 ◽  
Vol 1 (4) ◽  
pp. 150
Author(s):  
Veton Zejnullahi
Keyword(s):  
New Technology ◽  
World Order ◽  
Modern Society ◽  
Media Access ◽  
Four Corners ◽  
The World ◽  
The Media ◽  
The Republic ◽  
Human Eyes ◽  
The Impact

The process of globalization, which many times is considered as new world order is affecting all spheres of modern society but also the media. In this paper specifically we will see the impact of globalization because we see changing the media access to global problems in general being listed on these processes. We will see that the greatest difficulties will have small media as such because the process is moving in the direction of creating mega media which thanks to new technology are reaching to deliver news and information at the time of their occurrence through choked the small media. So it is fair to conclude that the rapid economic development and especially the technology have made the world seem "too small" to the human eyes, because for real-time we will communicate with the world with the only one Internet connection, and also all the information are take for the development of events in the four corners of the world and direct from the places when the events happen. Even Albanian space has not left out of this process because the media in the Republic of Albania and the Republic of Kosovo are adapted to the new conditions under the influence of the globalization process. This fact is proven powerful through creating new television packages, written the websites and newspapers in their possession.

Some aspects of familiarization with binomial coefficients

2019 ◽  
pp. 49-53
Author(s):  
Abdulkarim Magomedov ◽  
S.A. Lavrenchenko
Keyword(s):  
Elementary Mathematics ◽  
Binomial Coefficients ◽  
Computational Aspects

New laconic proofs of two classical statements of combinatorics are proposed, computational aspects of binomial coefficients are considered, and examples of their application to problems of elementary mathematics are given.

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