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Journal articles on the topic 'Combinatorial identities'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Sachdeva, Rachna, and Ashok Kumar Agarwal. "Further Rogers-Ramanujan type identities for modified lattice paths." Contributions to Discrete Mathematics 18, no. 2 (2023): 74–90. http://dx.doi.org/10.55016/ojs/cdm.v18i2.73702.

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Recently, the authors introduced the modified lattice paths which generalize Agarwal-Bressoud weighted lattice paths. Using these new objects they interpreted combinatorially two basic series identities which led to two new combinatorial Rogers-Ramanujan type identities. In this paper we obtain three more Rogers-Ramanujan type identities for modified lattice paths. This also leads to three new 3-way combinatorial identities.

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2

Lockwood, Elise, Zackery Reed, and Sarah Erickson. "Undergraduate Students’ Combinatorial Proof of Binomial Identities." Journal for Research in Mathematics Education 52, no. 5 (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

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3

Morgan, Thomas L. "Six Combinatorial Identities." SIAM Review 30, no. 2 (1988): 308–9. http://dx.doi.org/10.1137/1030055.

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4

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

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5

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

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6

Marwah, Bhanu, and Megha Goyal. "Split lattice paths and Rogers-Ramanujan type identities." Contributions to Discrete Mathematics 19, no. 3 (2024): 241–57. http://dx.doi.org/10.55016/ojs/cdm.v19i3.75377.

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In this paper, an open problem posed by the second author [On $q$-series and split lattice paths, Graphs and Combinatorics, 2020] is addressed. Here, we provide combinatorial interpretations of four generalized basic series in terms of split lattice paths. Out of these series, two series have been studied by Adiga et. al. [On Generalization of Some Combinatorial Identities, J. Ramanujan Soc. of Math. and Math. Sc., 2016] using split $(n + t)$-color partitions and $R$-weighted lattice paths but a direct one-to-one correspondence between these two classes was missing. We are successful in the qu

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7

Silva, Reginaldo Leoncio, and Elen Viviani Pereira Spreafico. "ON COMBINATORIAL IDENTITIES FOR R-GENERALIZED FIBONACCI SEQUENCES." Revista Sergipana de Matemática e Educação Matemática 9, no. 3 (2024): 124–35. http://dx.doi.org/10.34179/revisem.v9i3.21331.

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In this paper, we investigate combinatorial identities for r−generalized Fibonacci sequences. For this purpose, we established a combinatorial fundamental system related to the sequences of r−generalized Fibonacci type, and using the properties of the Casoratian matrix associated we obtain new combinatorial identities. Moreover, some special cases are studied and new general combinatorial identities are provided for these special sequences of numbers. Keywords: Fundamental System, Properties, Combinatorial Identities.

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8

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

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9

Lavertu, Marie-Louis, and Claude Levesque. "On Bernstein's Combinatorial Identities." Fibonacci Quarterly 23, no. 4 (1985): 347–55. http://dx.doi.org/10.1080/00150517.1985.12429805.

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10

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

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11

Annamalai, Chinnaraji. "Computation and Calculus for Combinatorial Geometric Series and Binomial Identities and Expansions." Journal of Engineering and Exact Sciences 8, no. 7 (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 engineerin

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12

Sonik, Pooja, D. Ranganatha, and Megha Goyal. "On a generalized basic series and Rogers-Ramanujan type identities." Contributions to Discrete Mathematics 18, no. 1 (2023): 15–28. http://dx.doi.org/10.55016/ojs/cdm.v18i1.73025.

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In this paper, we give the generalization of MacMahon's type combinatorial identities. A generalized $q$-series is interpreted as the generating function of two different combinatorial objects, viz., restricted $n$-color partitions and weighted lattice paths which give entirely new Rogers–Ramanujan–MacMahon type combinatorial identities. This result yields an infinite class of 2-way combinatorial identities which further extends the work of Agarwal and Goyal. We also discuss the bijective proof of the main result. Forbye, eight particular cases are also discussed which give a combinatorial int

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13

U, Sung Sik, and Kyu Song Chae. "Proof of some combinatorial identities by an analytic method." Online Journal of Analytic Combinatorics, no. 16 (December 31, 2021): 1–11. https://doi.org/10.61091/ojac-1603.

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We prove some combinatorial identities by an analytic method. We use the property that singular integrals of particular functions include binomial coefficients. In this paper, we prove combinatorial identities from the fact that two results of the particular function calculated as two ways using the residue theorem in the complex function theory are the same. These combinatorial identities are the generalization of a combinatorial identity that has been already obtained

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14

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 applicat

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15

Wenchang, Chu. "Inversion techniques and combinatorial identities. Basic hypergeometric identities." Publicationes Mathematicae Debrecen 44, no. 3-4 (1994): 301–20. http://dx.doi.org/10.5486/pmd.1994.1367.

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16

MEURMAN, ARNE, and MIRKO PRIMC. "A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE." Communications in Contemporary Mathematics 03, no. 04 (2001): 593–614. http://dx.doi.org/10.1142/s0219199701000512.

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J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper

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17

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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18

Benjamin, Arthur T., and Elizabeth Reiland. "Combinatorial Proofs of Fibonomial Identities." Fibonacci Quarterly 52, no. 5 (2014): 28–34. http://dx.doi.org/10.1080/00150517.2014.12427854.

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19

Thu, T. D. "Identities for Combinatorial Extremal Theory." Bulletin of the London Mathematical Society 29, no. 6 (1997): 693–96. http://dx.doi.org/10.1112/s0024609397003238.

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20

Neto, Antônio Francisco, and Petrus H. R. dos Anjos. "Zeon Algebra and Combinatorial Identities." SIAM Review 56, no. 2 (2014): 353–70. http://dx.doi.org/10.1137/130906684.

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21

Wilf, Herbert S., and Doron Zeilberger. "Rational functions certify combinatorial identities." Journal of the American Mathematical Society 3, no. 1 (1990): 147. http://dx.doi.org/10.1090/s0894-0347-1990-1007910-7.

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22

Sun, Zhi-Wei. "Combinatorial identities in dual sequences." European Journal of Combinatorics 24, no. 6 (2003): 709–18. http://dx.doi.org/10.1016/s0195-6698(03)00062-3.

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23

Tsylova, E. G. "One class of combinatorial identities." Journal of Soviet Mathematics 39, no. 2 (1987): 2672–78. http://dx.doi.org/10.1007/bf01084978.

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24

Tsylova, E. G. "Combinatorial identities and Polya walks." Journal of Soviet Mathematics 40, no. 2 (1988): 247–50. http://dx.doi.org/10.1007/bf01085120.

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25

Ismail, Mourad E. H., and Dennis Stanton. "Some Combinatorial and Analytical Identities." Annals of Combinatorics 16, no. 4 (2012): 755–71. http://dx.doi.org/10.1007/s00026-012-0158-1.

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26

Schneider, R. "Combinatorial identities for polyhedral cones." St. Petersburg Mathematical Journal 29, no. 1 (2017): 209–21. http://dx.doi.org/10.1090/spmj/1489.

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27

Deng, Yingpu. "A class of combinatorial identities." Discrete Mathematics 306, no. 18 (2006): 2234–40. http://dx.doi.org/10.1016/j.disc.2006.04.021.

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28

Hamel, A. M. "Pfaffian Identities: A Combinatorial Approach." Journal of Combinatorial Theory, Series A 94, no. 2 (2001): 205–17. http://dx.doi.org/10.1006/jcta.2000.3117.

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29

Verde-Star, Luis. "Divided Differences and Combinatorial Identities." Studies in Applied Mathematics 85, no. 3 (1991): 215–42. http://dx.doi.org/10.1002/sapm1991853215.

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30

Chen, Yulei, and Dongwei Guo. "Combinatorial Identities Concerning Binomial Quotients." Symmetry 16, no. 6 (2024): 746. http://dx.doi.org/10.3390/sym16060746.

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Making use of a telescoping approach, three types of sums of binomial quotients are examined. The summation terms of the two types of alternating sums have symmetry (i.e., their numerators and denominators are completely symmetric). We obtained a series of their explicit sums. Furthermore, by means of binomial relations, three recurrence relations of the sums are derived. In addition, series of double summation formulae involving binomial quotients are established.

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31

Sthanumoorthy, Neelacanta, and Kandasamy Priyadharsini. "Root supermultiplicities and corresponding combinatorial identities for some Borcherds superalgebras." Glasnik Matematicki 49, no. 1 (2014): 53–81. http://dx.doi.org/10.3336/gm.49.1.06.

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32

KIM, BYUNGCHAN. "COMBINATORIAL PROOFS OF CERTAIN IDENTITIES INVOLVING PARTIAL THETA FUNCTIONS." International Journal of Number Theory 06, no. 02 (2010): 449–60. http://dx.doi.org/10.1142/s1793042110003046.

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In this brief note, we give combinatorial proofs of two identities involving partial theta functions. As an application, we prove an identity for the product of partial theta functions, first established by Andrews and Warnaar. We also provide a generalization of the first two identities and give a combinatorial proof of the generalized identities.

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33

Méndez, Miguel A., and José L. Ramírez. "A new approach to the r-Whitney numbers by using combinatorial differential calculus." Acta Universitatis Sapientiae, Mathematica 11, no. 2 (2019): 387–418. http://dx.doi.org/10.2478/ausm-2019-0029.

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Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several kn

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34

Chaudhary, M. P. "Relations between Rα, Rβ and Rm functions related to Jacobi’s triple-product identity and the family of theta-function identities". Notes on Number Theory and Discrete Mathematics 27, № 2 (2021): 1–11. http://dx.doi.org/10.7546/nntdm.2021.27.2.1-11.

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In this paper, the author establishes a set of three new theta-function identities involving Rα, Rβ and Rm functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of Rα, Rβ and Rm (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some clos

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35

Chaudhary, M. P., and A. Vanitha. "CERTAIN IDENTITIES ASSOCIATED WITH EISENSTEIN SERIES, G¨ OLLNITZ-GORDON IDENTITIES AND COMBINATORIAL PARTITION IDENTITIES FOR THE CONTINUED FRACTIONS OF ORDER SIXTEEN." JOURNAL OF RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES 11, no. 02 (2022): 55–62. http://dx.doi.org/10.56827/jrsmms.2024.1102.3.

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The objective of this paper is to establish six new identities which depict interrelationships between Eisenstein series identities, G¨ ollnitz-Gordon identities and combinatorial partition identities.

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36

Adegoke, Kunle, and Robert Frontczak. "Some notes on an identity of Frisch." Open Journal of Mathematical Sciences 8 (December 31, 2024): 216–26. https://doi.org/10.30538/oms2024.0237.

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In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which can be inferred straightforwardly from our results.

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37

Guo, Victor. "On Jensen's and related combinatorial identities." Applicable Analysis and Discrete Mathematics 5, no. 2 (2011): 201–11. http://dx.doi.org/10.2298/aadm110717017g.

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Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity n?k=0 (x+kz/k)(y-kz/n-k = n?k=0 (x+y-k)/n-k)zk; and Chu's and Mohanty-Handa's generalizations of Jensen's identity. We also give a quite simple proof of an equivalent form of Graham-Knuth-Patashnik's identity ?k?0 (m+r/m-n-k) (n+k/n)xm-n-k yk = ?k?0 (-r/m-n-k)(n+k/n)(-x)m-n-k(x+y)k; which was rediscovered, respectively, by Sun in 2003 and Munarini in 2005. Finally we give a multinomial coefficient generalization of this identity.

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38

Bounebirat, Fouad, Diffalah Laissaoui, and Mourad Rahmani. "Some combinatorial identities via Stirling transform." Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 92–98. http://dx.doi.org/10.7546/nntdm.2018.24.4.92-98.

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39

Bianconi, Ryan, Marcus Elia, Akalu Tefera, and Aklilu Zeleke. "On proofs of certain combinatorial identities." Involve, a Journal of Mathematics 14, no. 4 (2021): 697–702. http://dx.doi.org/10.2140/involve.2021.14.697.

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40

郑, 欢欢. "Several Methods for Proving Combinatorial Identities." Pure Mathematics 11, no. 06 (2021): 1137–45. http://dx.doi.org/10.12677/pm.2021.116128.

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41

Plaza, A., and S. Falcón. "Combinatorial proofs of Honsberger-type identities." International Journal of Mathematical Education in Science and Technology 39, no. 6 (2008): 785–92. http://dx.doi.org/10.1080/00207390801986916.

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42

Ewell, John A. "Some combinatorial identities and arithmetical applications." Rocky Mountain Journal of Mathematics 15, no. 2 (1985): 365–70. http://dx.doi.org/10.1216/rmj-1985-15-2-365.

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43

Soshnikov, Alexander. "compact groups and related combinatorial identities." Annals of Probability 28, no. 3 (2000): 1353–70. http://dx.doi.org/10.1214/aop/1019160338.

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44

Gerdemann, Dale. "Combinatorial Proofs of Zeckendorf Family Identities." Fibonacci Quarterly 46-47, no. 3 (2008): 249–61. http://dx.doi.org/10.1080/00150517.2008.12428159.

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45

Bramham, Alex, and Martin Griffiths. "Combinatorial Interpretations of Some Convolution Identities." Fibonacci Quarterly 54, no. 4 (2016): 335–39. http://dx.doi.org/10.1080/00150517.2016.12427801.

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46

Regev, Amitai. "$S_{\infty }$ representations and combinatorial identities." Transactions of the American Mathematical Society 353, no. 11 (2001): 4371–404. http://dx.doi.org/10.1090/s0002-9947-01-02772-6.

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47

Chamberland, Marc. "Factored matrices can generate combinatorial identities." Linear Algebra and its Applications 438, no. 4 (2013): 1667–77. http://dx.doi.org/10.1016/j.laa.2011.08.030.

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48

Lee, Gwang-Yeon, Jin-Soo Kim, and Seong-Hoon Cho. "Some combinatorial identities via Fibonacci numbers." Discrete Applied Mathematics 130, no. 3 (2003): 527–34. http://dx.doi.org/10.1016/s0166-218x(03)00331-7.

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49

Strehl, Volker. "Binomial identities — combinatorial and algorithmic aspects." Discrete Mathematics 136, no. 1-3 (1994): 309–46. http://dx.doi.org/10.1016/0012-365x(94)00118-3.

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50

Joichi, J. T. "Hecke–Rogers, Andrews identities; combinatorial proofs." Discrete Mathematics 84, no. 3 (1990): 255–59. http://dx.doi.org/10.1016/0012-365x(90)90131-z.

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