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Dissertations / Theses on the topic 'Combinatorial identities'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Reiland, Elizabeth. "Combinatorial Interpretations of Fibonomial Identities." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/10.

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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

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2

Preston, Greg. "Combinatorial Explanations of Known Harmonic Identities." Scholarship @ Claremont, 2001. https://scholarship.claremont.edu/hmc_theses/134.

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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.

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3

Dohmen, Klaus. "Improved Bonferroni inequalities via abstract tubes : inequalities and identities of inclusion-exclusion type /." Berlin [u.a.] : Springer, 2003. http://www.loc.gov/catdir/enhancements/fy0818/2003066695-d.html.

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4

Heberle, Curtis. "A Combinatorial Approach to $r$-Fibonacci Numbers." Scholarship @ Claremont, 2012. https://scholarship.claremont.edu/hmc_theses/34.

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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.

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5

Ryoo, Ji Hoon. "Identities for the Multiple Polylogarithm Using the Shuffle Operation." Fogler Library, University of Maine, 2001. http://www.library.umaine.edu/theses/pdf/RyooJH2001.pdf.

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6

Silva, Robson da. "Provas bijetivas atraves de nova representação matricial para partições." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307496.

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Orientador: Jose Plinio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-14T00:20:04Z (GMT). No. of bitstreams: 1 Silva_Robsonda_D.pdf: 897208 bytes, checksum: 5d17d33a20271484f3f7853e008443db (MD5) Previous issue date: 2009<br>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 conseq

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7

Ribeiro, Andreia Cristina. "Aspectos combinatorios de identidades do tipo Rogers-Ramanujan." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307502.

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Orientador: Jose Plinio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-07T19:25:43Z (GMT). No. of bitstreams: 1 Ribeiro_AndreiaCristina_D.pdf: 576297 bytes, checksum: 445154b7e26e801e909854c976d31c45 (MD5) Previous issue date: 2006<br>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

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8

Alegri, Mateus. "Interpretações combinatórias para identidades envolvendo sobrepartições e partições planas." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307516.

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Orientador: José Plínio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisitca e Computação Cientifica<br>Made available in DSpace on 2018-08-16T01:34:00Z (GMT). No. of bitstreams: 1 Alegri_Mateus_D.pdf: 32503931 bytes, checksum: fb4329080c2c9c80896a52e4442b1b86 (MD5) Previous issue date: 2010<br>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

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9

Cruz, Carla Maria. "Numerical and combinatorial applications of generalized Appell polynomials." Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/13962.

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Doutoramento em Matemática<br>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 constructi

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10

Shibalovich, Paul. "Fundamental theorem of algebra." CSUSB ScholarWorks, 2002. https://scholarworks.lib.csusb.edu/etd-project/2203.

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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.

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11

Gioev, Dimitri. "Generalizations of Szego Limit Theorem : Higher Order Terms and Discontinuous Symbols." Doctoral thesis, KTH, Mathematics, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3123.

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12

Konvalinka, Matjaž. "Combinatorics of determinantal identities." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43790.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.<br>This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>Includes bibliographical references (p. 125-129).<br>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 th

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13

Spreafico, Elen Viviani Pereira 1986. "Novas identidades envolvendo os números de Fibonacci, Lucas e Jacobsthal via ladrilhamentos." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307509.

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Orientador: José Plínio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica<br>Made available in DSpace on 2018-08-26T02:14:38Z (GMT). No. of bitstreams: 1 Spreafico_ElenVivianiPereira_D.pdf: 1192138 bytes, checksum: 2b12cd351b94a0f2f7ec24fc172305c9 (MD5) Previous issue date: 2014<br>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 demo

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14

Mucelin, Cláudio. "Demonstrações bijetivas em partições." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306031.

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Orientador: Andréia Cristina Ribeiro<br>Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-17T16:44:00Z (GMT). No. of bitstreams: 1 Mucelin_Claudio_M.pdf: 744549 bytes, checksum: 062211ac0a3abf9bcf171fe9881dcafa (MD5) Previous issue date: 2011<br>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,

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15

Afsharijoo, Pooneh. "Looking for a new version of Gordon's identities : from algebraic geometry to combinatorics through partitions." Thesis, Sorbonne Paris Cité, 2019. https://theses.md.univ-paris-diderot.fr/AFSHARIJOO_Pooneh_2_complete_20190510.pdf.

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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 ide

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16

Bennett, Robert. "Fibonomial Tilings and Other Up-Down Tilings." Scholarship @ Claremont, 2016. https://scholarship.claremont.edu/hmc_theses/84.

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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 combin

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17

Huang, Yen-Jung, and 黃嬿蓉. "Combinatorial Identities from Lagrange's Interpolation Polynomial." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/30250069414157573515.

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碩士<br>國立交通大學<br>應用數學系所<br>103<br>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.

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18

Chung, Chan-Liang, and 鍾展良. "On Generalized Combinatorial Identities among Multiple Zeta Values." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/40844420096722366223.

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博士<br>國立中正大學<br>數學研究所<br>101<br>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<n_2<...<n_q} n_1^(s_1) n_2^(s_2) … n_q^(s_q). In this thesis, we introduce a shuffle product process of two multiple zeta values and obtain some shuffle relations. Also, we obtain important combinatorial identities among multiple zeta values and beyond their shuffle relations. We also discuss the classical Vandermonde's convolution and give our version of convolution-type identity such as Σ_{a+b=k} (m+|a|)choose(m) M(a)M(b) (n+

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19

Yen, Miau-Hua, and 顏妙樺. "Combinatorial Identities of Convolution Type through Shuffle Products." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/26112638583487325303.

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碩士<br>國立中正大學<br>應用數學研究所<br>102<br>For an $r$-tuple of positive integers $\alpha_1, \alpha_2, \ldots , \alpha_r$ with $\alpha_r \geq 2$ ,the multiple zeta value or $r$-fold Euler sum $\zeta(\boldsymbol{\alpha})$ is defined as \[ \zeta(\boldsymbol{\alpha}) = \sum_{1 \leq n_1 < n_2 < \cdots <n_r} n_1^{- \alpha_1} n_2^{- \alpha_2} \cdots n_r^{- \alpha_r}, \] the numbers $\vert \boldsymbol{\alpha} \vert = \alpha_1 + \alpha_2 + \cdots + \alpha_r$ is called the weight of $\zeta(\alpha_1, \alpha_2, \ldots , \alpha_r)$ and $r$ is its depth. In this thesis, we shall use shuffle products of multiple

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20

Cook, William J. "Affine lie algebras, vertex operator algebras and combinatorial identities." 2005. http://www.lib.ncsu.edu/theses/available/etd-03232005-234709/unrestricted/etd.pdf.

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21

Tsai, Shing-Jing, and 蔡幸靜. "Combinatorial Identities from the Generating Functions of Two Binomial Coefficients." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/7uh3gp.

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碩士<br>國立中正大學<br>數學研究所<br>102<br>In recent years, it is interesting to know more about the multiple zeta values. In this thesis, we shall take a look at some histories about the multiple zeta values in Section 1. In Section 2, we focus on the Drinfel’d integral and its applications. Drinfel’d integral is a very useful representation for a multiple zeta value, so that we can express a multiple zeta value as an integral of a string of differential forms of two simple types instead of summation forms. In particular, the number of variables in the integral can be reduced. Also, there are many way

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22

Campbell, Geoffrey Bruce. "Combinatorial identities in number theory related to q-series and arithmetical functions." Phd thesis, 1997. http://hdl.handle.net/1885/145356.

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23

Hong, Yi-Hui, and 洪宜慧. "Some Combinatorial Identities Produced from Shuffle Products of Two Sums of Multiple Zeta Values." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/38125669081032208248.

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碩士<br>國立中正大學<br>數學研究所<br>101<br>Abstract A multiple zeta value of depth r and weight j with a multi-index = (1; 2; : : : ; r) of positive integers with r 2. After the shuffle process product of a particular multiple zeta value of weight w1 and the sum of the multiple zeta value of weight w2. It will produce some combinatorial identities of convolution type.

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24

Nyirenda, Darlison. "Analytic and combinatorial explorations of partitions associated with the Rogers-Ramanujan identities and partitions with initial repetitions." Thesis, 2016. http://hdl.handle.net/10539/21040.

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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016.<br>In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partit

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