Relevant bibliographies by topics / Combinatorial identities

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Combinatorial identities'

Author: Grafiati
Published: 4 June 2021
Last updated: 30 January 2023

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Journal articles on the topic "Combinatorial identities"

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Lockwood, Elise, Zackery Reed, and Sarah Erickson. "Undergraduate Students’ Combinatorial Proof of Binomial Identities." Journal for Research in Mathematics Education 52, no. 5 (November 2021): 539–80. http://dx.doi.org/10.5951/jresematheduc-2021-0112.

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Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.
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Morgan, Thomas L. "Six Combinatorial Identities." SIAM Review 30, no. 2 (June 1988): 308–9. http://dx.doi.org/10.1137/1030055.

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Morgan, Thomas L. "Six Combinatorial Identities." SIAM Review 31, no. 2 (June 1989): 325–28. http://dx.doi.org/10.1137/1031063.

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Xin-Rong, Ma, and Wang Tian-Ming. "Two Combinatorial Identities." SIAM Review 37, no. 1 (March 1995): 98. http://dx.doi.org/10.1137/1037009.

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Mestechkin, M. "On two combinatorial identities." Journal of Computational Methods in Sciences and Engineering 17, no. 4 (November 24, 2017): 887–912. http://dx.doi.org/10.3233/jcm-170763.

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Hernández-Galeana, A., Elizabeth Santiago-Cort´es, and Jose Luis López Bonilla. "On certain combinatorial identities." Journal de Ciencia e Ingeniería 14, no. 1 (June 29, 2022): 34–38. http://dx.doi.org/10.46571/jci.2022.1.4.

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Annamalai, Chinnaraji. "Computation and Calculus for Combinatorial Geometric Series and Binomial Identities and Expansions." Journal of Engineering and Exact Sciences 8, no. 7 (September 22, 2022): 14648–01. http://dx.doi.org/10.18540/jcecvl8iss7pp14648-01i.

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Nowadays, the growing complexity of mathematical and computational modelling demands the simplicity of mathematical and computational equations for solving today’s scientific problems and challenges. This paper presents combinatorial geometric series, innovative binomial coefficients, combinatorial equations, binomial expansions, calculus with combinatorial geometric series, and innovative binomial theorems. Combinatorics involves integers, factorials, binomial coefficients, discrete mathematics, and theoretical computer science for finding solutions to the problems in computing and engineering science. The combinatorial geometric series with binomial expansions and its theorems refer to the methodological advances which are useful for researchers who are working in computational science. Computational science is a rapidly growing multi-and inter-disciplinary area where science, engineering, computation, mathematics, and collaboration use advance computing capabilities to understand and solve the most complex real-life problems.
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Wenchang, Chu. "Inversion techniques and combinatorial identities. Basic hypergeometric identities." Publicationes Mathematicae Debrecen 44, no. 3-4 (April 1, 1994): 301–20. http://dx.doi.org/10.5486/pmd.1994.1367.

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Chabaud, Ulysse, Abhinav Deshpande, and Saeed Mehraban. "Quantum-inspired permanent identities." Quantum 6 (December 19, 2022): 877. http://dx.doi.org/10.22331/q-2022-12-19-877.

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The permanent is pivotal to both complexity theory and combinatorics. In quantum computing, the permanent appears in the expression of output amplitudes of linear optical computations, such as in the Boson Sampling model. Taking advantage of this connection, we give quantum-inspired proofs of many existing as well as new remarkable permanent identities. Most notably, we give a quantum-inspired proof of the MacMahon master theorem as well as proofs for new generalizations of this theorem. Previous proofs of this theorem used completely different ideas. Beyond their purely combinatorial applications, our results demonstrate the classical hardness of exact and approximate sampling of linear optical quantum computations with input cat states.
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Munarini, Emanuele. "Combinatorial identities for Appell polynomials." Applicable Analysis and Discrete Mathematics 12, no. 2 (2018): 362–88. http://dx.doi.org/10.2298/aadm161001004m.

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Using the techniques of the modern umbral calculus, we derive several combinatorial identities involving s-Appell polynomials. In particular, we obtain identities for classical polynomials, such as the Hermite, Laguerre, Bernoulli, Euler, N?rlund, hypergeometric Bernoulli, and Legendre polynomials. Moreover, we obtain a generalization of Carlitz's identity for Bernoulli numbers and polynomials to arbitrary symmetric s-Appell polynomials.
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Dissertations / Theses on the topic "Combinatorial identities"

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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 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.
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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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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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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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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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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
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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
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
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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
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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
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
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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
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisitca e Computação Cientifica
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
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
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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
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.
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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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Books on the topic "Combinatorial identities"

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1931-, Wilf Herbert S., and Zeilberger Doron, eds. A=B. Wellesley, Mass: A K Peters, 1996.

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Meurman, Arne. Annihilating fields of standard modules of Sl(2, C) ãnd combinatorial identities. Providence, RI: American Mathematical Society, 1999.

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Improved Bonferroni inequalities via abstract tubes: Inequalities and identities of inclusion-exclusion type. Berlin: Springer-Verlag, 2003.

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An invitation to q-series: From Jacobi's triple product identity to Ramanujan's "most beautiful identity". Singapore: World Scientific Pub Co., 2011.

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Farkas, Hershel M. Theta constants, Riemann surfaces, and the modular group: An introduction with applications to uniformization theorems, partition identities, and combinatorial number theory. Providence, R.I: American Mathematical Society, 2001.

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Flannery, D. L. (Dane Laurence), 1965-, ed. Algebraic design theory. Providence, R.I: American Mathematical Society, 2011.

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Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. Providence, Rhode Island: American Mathematical Society, 2014.

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Jacques, Sauloy, and Singer Michael F. 1950-, eds. Galois theories of linear difference equations: An introduction. Providence, Rhode Island: American Mathematical Society, 2016.

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Giambruno, Antonio, Amitai Regev, and Mikhail Zaicev. Polynomial Identities and Combinatorial Methods. Taylor & Francis Group, 2003.

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Giambruno, Antonio, Amitai Regev, and Mikhail Zaicev. Polynomial Identities and Combinatorial Methods. Taylor & Francis Group, 2003.

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Book chapters on the topic "Combinatorial identities"

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Chu, Wenchang. "Inversion Techniques and Combinatorial Identities." In Runs and Patterns in Probability, 31–57. Boston, MA: Springer US, 1994. http://dx.doi.org/10.1007/978-1-4613-3635-8_3.

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Erusalimsky, Ya M. "Combinatorial Identities with Binomial Coefficients." In Operator Theory and Harmonic Analysis, 135–46. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76829-4_6.

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Wei, Chuanan, and Ling Wang. "New Proofs for Several Combinatorial Identities." In Communications in Computer and Information Science, 30–39. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-34289-9_4.

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Roytvarf, Alexander A. "Jacobi Identities and Related Combinatorial Formulas." In Thinking in Problems, 1–23. Boston: Birkhäuser Boston, 2012. http://dx.doi.org/10.1007/978-0-8176-8406-8_1.

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Nathanson, Melvyn B. "Adjoining Identities and Zeros to Semigroups." In Combinatorial and Additive Number Theory, 195–201. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-1601-6_14.

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Alzer, Horst, Omran Kouba, and Man Kam Kwong. "Combinatorial Identities and Inequalities for Trigonometric Sums." In Springer Optimization and Its Applications, 7–33. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-55857-4_2.

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Andersen, Erik Sparre, and Mogens Esrom Larsen. "Combinatorial Identities: A Generalization of Dougall’s Identity." In Advances in Combinatorial Methods and Applications to Probability and Statistics, 77–88. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-1-4612-4140-9_5.

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Croot, Ernest. "A combinatorial method for developing Lucas sequence identities." In CRM Proceedings and Lecture Notes, 175–78. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/crmp/046/13.

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Baker, Andrew. "Combinatorial and arithmetic identities based on formal group laws." In Algebraic Topology Barcelona 1986, 17–34. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0082998.

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Zubrilin, K. A. "Combinatorial Aspects of Capelli Identities and Structure of Algebras." In Formal Power Series and Algebraic Combinatorics, 785–88. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-662-04166-6_77.

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Conference papers on the topic "Combinatorial identities"

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Yang, Harold R. L. "Combinatorial proofs of Fibonacci identities." In 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2022), edited by Chi-Hua Chen and Hari Mohan Srivastava. SPIE, 2022. http://dx.doi.org/10.1117/12.2639169.

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Svrtan, Dragutin. "On some partition identities related to affine Lie algebra representations." In 1st Croatian Combinatorial Days. University of Zagreb Faculty of Civil Engineering, 2017. http://dx.doi.org/10.5592/co/ccd.2016.06.

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Damiano, Alberto, Vladimír Souček, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "Dirac Operator in Several Variables and Combinatorial Identities." In Numerical Analysis and Applied Mathematics. AIP, 2007. http://dx.doi.org/10.1063/1.2790256.

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Yuluklu, Eda. "Identities for Hermite base combinatorial polynomials and numbers." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2019. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0031017.

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Cação, I., M. I. Falcão, and H. R. Malonek. "Combinatorial identities in the context of hypercomplex function theory." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2017). Author(s), 2018. http://dx.doi.org/10.1063/1.5043904.

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Kasparian, Azniv. "Riemann-Roch Theorem and Mac Williams identities for an additive code with respect to a saturated lattice." In 2020 Algebraic and Combinatorial Coding Theory (ACCT). IEEE, 2020. http://dx.doi.org/10.1109/acct51235.2020.9383243.

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