Dissertations / Theses on the topic 'Combinatorial identities'

The Fibonomial numbers are defined by \[ \begin{bmatrix}n \\ k \end{bmatrix} = \frac{\prod_{i=n-k+1} ^{n} F_i}{\prod_{j=1}^{k} F_j} \] where $F_i$ is the $i$th Fibonacci number, defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0,F_1=1$. In the past year, Sagan and Savage have derived a combinatorial interpretation for these Fibonomial numbers, an interpretation that relies upon tilings of a partition and its complement in a given grid.In this thesis, I investigate previously proven theorems for the Fibonomial numbers and attempt to reinterpret and reprove them in light of this new combinatorial description. I also present combinatorial proofs for some identities I did not find elsewhere in my research and begin the process of creating a general mapping between the two different Fibonomial interpretations. Finally, I provide a discussion of potential directions for future work in this area.

We seek to discover combinatorial explanations of known identities involving harmonic numbers. Harmonic numbers do not readily lend themselves to combinatorial interpretation, since they are sums of fractions, and combinatorial arguments involve counting whole objects. It turns out that we can transform these harmonic identities into new identities involving Stirling numbers, which are much more apt to combinatorial interpretation. We have proved four of these identities, the first two being special cases of the third.

In this paper we explore generalized “$r$-Fibonacci Numbers” using a combinatorial “tiling” interpretation. This approach allows us to provide simple, intuitive proofs to several identities involving $r$-Fibonacci Numbers presented by F.T. Howard and Curtis Cooper in the August, 2011, issue of the Fibonacci Quarterly. We also explore a connection between the generalized Fibonacci numbers and a generalized form of binomial coefficients.

Orientador: Jose Plinio de Oliveira Santos
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: No presente trabalho, apresentamos provas bijetivas para algumas identidades. A principal ferramenta utilizada _e a representação para partições como matrizes de duas linhas introduzida em [9] e [10]. Também apresentamos algumas conseqüências desta representação e a extendemos a outros casos. Uma prova bijetiva para uma identidade envolvendo os Números Triangulares e apresentada ao final.
Abstract: In this work, we show bijective proofs for some identities. The main tool is the two-line matrix representation for partitions introduced in [9] and [10]. We also present some consequences of this representation and we also extend it to other cases. A bijective proof for an identity involving the Triangular Numbers is given at the end.
Doutorado
Matematica Discreta
Doutor em Matemática Aplicada

Orientador: Jose Plinio de Oliveira Santos
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho são estudadas varias das identidades do tipo Rogers-Ramanujan dadas por Slater. Em 1985, Andrews, introduziram um método geral para se estender para duas variáveis identidades desse tipo de modo a se obter, como casos especiais, certas importantes funções de Ramanujan. Santos, em 1991, forneceu conjecturas para varias das famílias de polinômios que surgem nestas extensões tendo provado algumas delas. Sills, em sua tese de doutorado, em 2002, implementou procedimentos que permitem a demonstra¸c¿ao das conjecturas dadas por Santos. No presente trabalho, de forma diferente daquela dada por Andrews, s¿ao introduzidos parâmetros nas somas que aparecem nestas identidades, de modo a se obter, em cada caso, funções geradoras que fornecem interpretações combinatórias para partições onde ¿números¿s¿ao vistos como ¿vetores¿e que fornecem, para especiais valores dos parâmetros, interpretações novas para muitas das identidades de Slater
Abstract: In this work many of the identities of the Rogers-Ramanujan type given by Slater are considered. In 1985, Andrews, introduced a general method in other to extend to two variables identities of this type in order to get, as special cases, some important functions of Ramanujan. Santos, in 1991, gave conjectures for many of the family of polynomials that appears in those extensions providing the proofs for some of them. Sills, in his Ph.D. thesis in 2002 ,has implemented procedures allowing the proofs of the conjectures given by Santos. In the present work, in a form different from the one given by Andrews, parameters are introduced in the sums of the identities in such a way to get, in each case, generating functions giving combinatorial interpretations for partitions where ¿numbers¿are represented as ¿vectors¿and that can give, as special cases, combinatorial interpretations for many of the identities given by Slater
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada

Orientador: José Plínio de Oliveira Santos
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisitca e Computação Cientifica
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Resumo: Neste trabalho apresentaremos novas provas bijetivas para identidades relacionadas a partições em partes pares e distintas, generalizações das identidades de Rogers-Ramanujan entre outras. Porém o objetivo principal será trabalhar com sobrepartições de inteiros, dando a estes uma nova interpretação em termos de matrizes de três linhas. Exibiremos provas bijetivas para algumas classes de sobrepartições, apresentaremos um novo resultado que basicamente é identificar uma sobrepartição com partições planas; sendo este o principal resultado deste trabalho. No final apresentaremos algumas aplicações da representação de partição via matrizes de duas linhas: fórmulas fechadas para algumas classes destas partições.
Abstract: In this work, we present new bijective proofs for identities related to partitions into distinct even parts, generalizations of Rogers-Ramanujan identities, among others. The basic aim is to work with overpartitions of integers, give a new interpretation in terms of three-line matrices. We will show bijective proofs for some classes of overpartitions. We will present a new result that is how to identify an overpartition (with some particularities) with plane partitions; which is one of the most important results. At the end we will present some applications of the representation of a partition as a two-line array: closed formulaes for some classes of these partitions.
Doutorado
Análise Combinatória
Doutor em Matemática Aplicada

Doutoramento em Matemática
This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.
Esta tese estuda propriedades e aplicações de diferentes polinómios de Appell generalizados no contexto da análise de Clifford. Exemplificando uma transformação realizada por polinómios de Appell generalizados, é introduzida uma transformação análoga à transformação de Joukowski complexa de ordem dois. A análise de um n- simplex de Pascal com entradas hipercomplexas permite sublinhar a relevância combinatória de polinómios hipercomplexos de Appell. O conceito de variáveis totalmente regulares e a sua relação com polinómios de Appell generalizados conduz à construção de novas bases para o espaço dos polinómios homogéneos holomorfos cujos elementos são todos isomorfos às potências inteiras da variável complexa. Por este motivo, tais polinómios são chamados de potências pseudo-complexas (PCP). Diferentes variantes de PCP são objeto de uma investigação detalhada. É dada especial atenção aos aspectos numéricos de PCP. Um algoritmo eficiente baseado em aritmética complexa é proposto para a sua implementação. Neste contexto, é apresentado um breve resumo de métodos numéricos para inverter matrizes de Vandermonde e é proposto um algoritmo modificado para ilustrar as vantagens de um tipo especial de PCP. Finalmente, são enfatizadas aplicações combinatórias de polinómios de Appell generalizados. A expressão explícita dos coeficientes de um tipo particular de polinómios de Appell e a sua relação com um simplex de Pascal com entradas hipercomplexas são obtidas. A comparação de dois tipos de polinómios de Appell tridimensionais leva à deteção de novas fórmulas envolvendo somas trigonométricas e de identidades combinatórias do tipo de Riordan – Sofo, caracterizadas pela sua expressão em termos de coeficientes binomiais centrais.

The fundamental theorem of algebra (FTA) is an important theorem in algebra. This theorem asserts that the complex field is algebracially closed. This thesis will include historical research of proofs of the fundamental theorem of algebra and provide information about the first proof given by Gauss of the theorem and the time when it was proved.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 125-129).
In this thesis, we apply combinatorial means for proving and generalizing classical determinantal identities. In Chapter 1, we present some historical background and discuss the algebraic framework we employ throughout the thesis. In Chapter 2, we construct a fundamental bijection between certain monomials that proves crucial for most of the results that follow. Chapter 3 studies the first, and possibly the best-known, determinantal identity, the matrix inverse formula, both in the commutative case and in some non-commutative settings (Cartier-Foata variables, right-quantum variables, and their weighted generalizations). We give linear-algebraic and (new) bijective proofs; the latter also give an extension of the Jacobi ratio theorem. Chapter 4 is dedicated to the celebrated MacMahon master theorem. We present numerous generalizations and applications. In Chapter 5, we study another important result, Sylvester's determinantal identity. We not only generalize it to non-commutative cases, we also find a surprising extension that also generalizes the master theorem. Chapter 6 has a slightly different, representation theory flavor; it involves representations of the symmetric group, and also Hecke algebras and their characters. We extend a result on immanants due to Goulden and Jackson to a quantum setting, and reprove certain combinatorial interpretations of the characters of Hecke algebras due to Ram and Remmel.
by Matjaž Konvalinka.
Ph.D.

Orientador: José Plínio de Oliveira Santos
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Neste trabalho, colaboramos com provas combinatórias que utilizam a contagem e a q-contagem de elementos em conjuntos de ladrilhamentos com restrições. Na primeira parte do trabalho utilizamos os ladrilhamentos para demonstrar algumas identidades da teoria das partições, dentre elas, o Teorema dos Números Triangulares e o Teorema q-análogo da Série q-Binomial. Na segunda parte do trabalho apresentamos interpretações combinatórias, via ladrilhamento, para algumas identidades envolvendo os números de Jacobsthal e os números generalizados de Jacobsthal . Na terceira parte do trabalho são dadas novas identidades envolvendo os números q-análogos de Jacobsthal e encontramos generalizações para essas novas identidades. Por fim, definimos duas novas sequências: números de Fibonacci generalizados e números de Lucas generalizados e, utilizando ladrilhamentos, estabelecemos e demonstramos novas identidades envolvendo esses números
Abstract: In this work we present combinatorial proofs by making use of tilings. In the first part we use tilings to prove some identities on Partitions Theory, including Triangular Numbers' Theorem and q-analogue of q-Binomial Theorem. In the second part we present combinatorial interpretations, using tilings, for some identities involving Jacobsthal numbers and generalized Jacobsthal numbers. Next we find new identities involving an q-analogue of Jacobsthal numbers and generalizations for these new identities. Finally, we define two new sequences: generalized Fibonacci numbers and generalized Lucas numbers, and using tilings, we prove new identities involving these numbers
Doutorado
Matematica Aplicada
Doutora em Matemática Aplicada

Orientador: Andréia Cristina Ribeiro
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Este trabalho apresenta alguns resultados sobre partições de números inteiros e a importância deles na história da Matemática e da Teoria dos Números. Encontrar demonstrações bijetivas em partições não é nada fácil. Mas, depois de encontradas, tornam-se uma maneira agradável e fácil de entender e provar algumas Identidades de Partições. Este trabalho pretende ser didático e de fácil entendimento para futuras pesquisas de estudantes que se interessem pelo assunto. Ele traz definições básicas e importantes sobre partições, os Gráficos de Ferrers, demonstrações de resultados interessantes como a Bijeção de Bressoud e o Teorema Pentagonal de Euler. Destaca também a importância das funções geradoras e alguns resultados devidos a Sylvester, Dyson, Fine, Schur e Rogers-Ramanujan
Abstract: This work presents some results about partitions of integers numbers and their importance in the history of Mathematics and in the Theory of the Numbers. To find bijective demonstrations in partitions it is not easy. But, after finding them, to understand and to prove some Identities of Partitions becomes agreeable and easy. This work intends to be didatic and of easy understanding for future researches made by students interested in this subject. It contains basic and important definitions about partitions, the Ferrers' Graphics, demonstrations of interesting results as the Bressond's Bijection and the Euler's Pentagonal Theorem. It also details the importance of the generating functions and some results due to Sylvester, Dyson, Fine, Schur and Rogers-Ramanujan
Mestrado
Teoria dos Numeros
Mestre em Matemática

Une partition d’un nombre entier positif n est une suite décroissante des entiers positifs dont la somme est égal à n. Les entiers qui y apparaissent sont appelés les parties de la partition. Ma thèse est centrée sur l’étude des partitions des nombres entiers et les identités qui les relient. Plus précisément, il s’agit de montrer que le nombre de partitions ayant une propriété A est égal au nombre de partitions ayant une autre propriété B. Ce type d’identité joue un rôle important en théorie des nombres, en combinatoire, en théorie de représentations et en physique statistique. Une de ces identités est la suivante: Théorème. (La première identité de Rogers-Ramanujan) Le nombre de partitions d’un nombre naturel n dont les parties sont congruentes à 1 ou 4 modulo 5 est égal au nombre de partitions de n dont les parties ne sont ni égales ni consécutives. Dans ce travail, on étudie les identités entre les partitions en utilisant la relation entre les combinatoires des partitions et les combinatoires de l’algèbre graduée associée à un objet important de la géométrie algèbrique: l’espace des arcs. Étant donnés un corps k de caractéristique zéro et des polynômes f1,…,fm de k[x1,…, xn], l’espace des arcs associé correspond à l’idéal I de S:=k[x1j,…, xnj|j>-1], engendré par les coefficients de certains développements associés aux polynômes ci-dessus et aux variables xij. Si on prend xi,0 = 0 pour i=1,…,n, la série de Hilbert-Poincaré de l’algèbre graduée S\I est étroitement liée aux partitions des entiers satisfaisants des conditions qui dépendent de l’idéal I. Dans le cas où f(x) = x^r de k[x], l’idéal I de k[x1, x2, … ] est un idéal différentiel pour la dérivation D(xi) = xi+1, dans le sens que DI est inclus dans I. En effet, dans ce cas I est engendré par x1^r et tous ses dérivés itérées. nous montrons que pour r = 2 le calcul de la base de Gröbner de l’ideal I par rapport à l’ordre lexicographique pondéré est lié à une identité faisant intervenir les partitions qui apparaissent dans la première identité de Rogers-Ramanujan. Nous prouvons ensuite qu’une base de Gröbner de cet idéal n’est pas différentiellement finie, au contraire du cas de l’ordre lexicographique inverse pondéré. Nous donnons une preuve de ce point de vue des identités de Gordon qui forment une famille importante d’identités reliant les partitions. En utilisant des idéaux différentiels et des méthodes venant de l’espace des arcs, nous énonçons une conjecture qui pourrait ajouter un nouveau membre aux identités de Gordon. Nous l’avons déjà démontré pour un cas particulier. À la fin, nous donnons une preuve simple et directe d’un théorème de Nguyen Duc Tam sur la base de Gröbner de l’idéal différentiel [x1y1]; Nous obtenons ensuite des identités entres les partitions avec 2 couleurs
A partition of a positive integer n is a decreasing sequence of positive integers such that their sum is equal to n. The integers which appear in this sequence are called the parts of this partition. My thesis studies the partitions of integers and the identities between them. A partition identity is an equality between the number of partitions of an integer n satisfying a certain condition A and the number of partitions of n satisfying another condition B. They play an important role in many areas: number theory, combinatorics, Lie theory, particle physics and statistical mechanics. One of these identities is as follows: Theorem. (The first Rogers-Ramanujan identity) The number of partitions of a positive integer n with no equal or consecutive parts is equal to the number of partitions of n into parts 1 or 4(mod.5). In this work, we study partition identities using the relation between the combinatorics of partitions and the combinatorics of graded algebras associated to an important object of algebraic geometry: arc spaces. Given a field k of characteristic zero and polynomials f1,…,fm in k[x1,…, xn], the associated arc space is the space corresponds to the ideal I of S:=k[x1j,…, xnj|j>-1], generated by the coefficients of some developments associated to the above polynomials and the variables xij. For focussed arcs, which is when we take xi0 = 0 for i=1,…,n, the Hilbert-Poincaré series of the graded algebra S\I is closely related to partitions of integers satisfying conditions depending on I. In the case where f(x) = x^r in k[x], the ideal I of k[x1, x2,… ] is a differential ideal for the derivation D(xi) = xi+1, in the sens that DI is included in I. In fact it is generated by x1^r and all its iterated derivatives. We show that when r = 2 the computation of a Gröbner basis of I with respect to the weighted lexicographical monomial order is related with an identity involving the partitions that appear in the first Rogers-Ramanujan identity. We then prove that a Gröbner basis of this ideal is not differentially finite in contrary with the case of the weighted reverse lexicographical order. We give a prove from this point of view of Gordon’s identities which is a family of important partitions identities. Using differential ideals we state a conjecture which could add a new member to Gordon’s identities. we prove then this conjecture for a special case. At the end, we give a simple and direct proof of a theorem of Nguyen Duc Tam about the Gröbner basis of the differential ideal [x1y1]; we then obtain identities involving partitions with 2 colors

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such as the Lucas numbers or the q-numbers, the total number of "tilings" is some multiple of a generalized binomial coefficient corresponding to the sequence chosen.

碩士
國立交通大學
應用數學系所
103
For a given real polynomial g(x) and infinite sequence a=(a_0,a_1,...) of distinct real numbers, we define the sequence L_a(g(x),n). We find that L_a(x^k,n) appears in coefficient of a term of some Lagrange's interpolation polynomial, and is also a generalization of the Stirling number of the second kind. Further properties of L_a(g(x),n) related to identities and combinatorial structure are given.

博士
國立中正大學
數學研究所
101
A multiple zeta value is defined for a string s=(s_1, s_2,..., s_q) of positive integers with s_q≧2 by ζ(s_1, s_2, ...,s_q)= Σ_{1≦n_1