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Journal articles on the topic 'Combinatorial identities'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 January 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

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

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12

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

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13

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

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14

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

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15

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

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16

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

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17

Ismail, Mourad E. H., and Dennis Stanton. "Some Combinatorial and Analytical Identities." Annals of Combinatorics 16, no. 4 (October 6, 2012): 755–71. http://dx.doi.org/10.1007/s00026-012-0158-1.

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18

Schneider, R. "Combinatorial identities for polyhedral cones." St. Petersburg Mathematical Journal 29, no. 1 (December 27, 2017): 209–21. http://dx.doi.org/10.1090/spmj/1489.

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19

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

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20

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

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21

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

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22

Sthanumoorthy, Neelacanta, and Kandasamy Priyadharsini. "Root supermultiplicities and corresponding combinatorial identities for some Borcherds superalgebras." Glasnik Matematicki 49, no. 1 (June 8, 2014): 53–81. http://dx.doi.org/10.3336/gm.49.1.06.

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23

MEURMAN, ARNE, and MIRKO PRIMC. "A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE." Communications in Contemporary Mathematics 03, no. 04 (November 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 we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

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24

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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25

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

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26

Bianconi, Ryan, Marcus Elia, Akalu Tefera, and Aklilu Zeleke. "On proofs of certain combinatorial identities." Involve, a Journal of Mathematics 14, no. 4 (October 23, 2021): 697–702. http://dx.doi.org/10.2140/involve.2021.14.697.

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27

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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28

Plaza, A., and S. Falcón. "Combinatorial proofs of Honsberger-type identities." International Journal of Mathematical Education in Science and Technology 39, no. 6 (September 15, 2008): 785–92. http://dx.doi.org/10.1080/00207390801986916.

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29

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

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30

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

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31

Regev, Amitai. "$S_{\infty }$ representations and combinatorial identities." Transactions of the American Mathematical Society 353, no. 11 (June 14, 2001): 4371–404. http://dx.doi.org/10.1090/s0002-9947-01-02772-6.

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32

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

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33

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

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34

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

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35

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

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36

Bera, Sudip, and Sajal Kumar Mukherjee. "Combinatorial proofs of some determinantal identities." Linear and Multilinear Algebra 66, no. 8 (August 29, 2017): 1659–67. http://dx.doi.org/10.1080/03081087.2017.1366970.

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37

Huang, I.-Chiau. "Applications of residues to combinatorial identities." Proceedings of the American Mathematical Society 125, no. 4 (1997): 1011–17. http://dx.doi.org/10.1090/s0002-9939-97-03923-3.

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38

Munagi, Augustine O. "Combinatorial identities for restricted set partitions." Discrete Mathematics 339, no. 4 (April 2016): 1306–14. http://dx.doi.org/10.1016/j.disc.2015.11.017.

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39

Xu, Xiaoping. "Skew-Symmetric Differential Operatorsand Combinatorial Identities." Monatshefte f�r Mathematik 127, no. 3 (April 22, 1999): 243–58. http://dx.doi.org/10.1007/s006050050037.

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40

Stanimirović, Stefan, Predrag Stanimirović, Marko Miladinović, and Aleksandar Ilić. "Catalan matrix and related combinatorial identities." Applied Mathematics and Computation 215, no. 2 (September 2009): 796–805. http://dx.doi.org/10.1016/j.amc.2009.06.003.

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41

Mansour, Toufik. "Combinatorial Identities and Inverse Binomial Coefficients." Advances in Applied Mathematics 28, no. 2 (February 2002): 196–202. http://dx.doi.org/10.1006/aama.2001.0774.

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42

Chapman, Robin. "Combinatorial Proofs of q-Series Identities." Journal of Combinatorial Theory, Series A 99, no. 1 (July 2002): 1–16. http://dx.doi.org/10.1006/jcta.2002.3251.

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43

Victor, Kowalenko. "Two methods for determining combinatorial identities." Annals of Mathematics and Physics 6, no. 1 (January 10, 2023): 007–11. http://dx.doi.org/10.17352/amp.000069.

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Two methods are presented for determining advanced combinatorial identities. The first is based on extending the original identity so that it can be expressed in terms of hypergeometric functions whereupon tabulated values of the functions can be used to reduce the identity to a simpler form. The second is a computer method based on Koepf's version of the Wilf-Zeilberger approach that has been implemented in a suite of intrinsic routines in Maple. As a consequence, some new identities are presented.

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44

Chaudhary, M. P., Sangeeta Chaudhary, and Sonajharia Minz. "On relationships between q-product identities and combinatorial partition identities." Mathematica Moravica 24, no. 1 (2020): 83–91. http://dx.doi.org/10.5937/matmor2001083c.

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45

KIM, BYUNGCHAN. "COMBINATORIAL PROOFS OF CERTAIN IDENTITIES INVOLVING PARTIAL THETA FUNCTIONS." International Journal of Number Theory 06, no. 02 (March 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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46

Chaudhary, M. P., Salahuddin, and Junesang Choi. "CERTAIN RELATIONSHIPS BETWEEN q-PRODUCT IDENTITIES, COMBINATORIAL PARTITION IDENTITIES AND CONTINUED-FRACTION IDENTITIES." Far East Journal of Mathematical Sciences (FJMS) 101, no. 5 (February 28, 2017): 973–82. http://dx.doi.org/10.17654/ms101050973.

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47

Chaudhary, Mahendra Pal, Getachew Abiye Salilew, and Junesang Choi. "FIVE RELATIONSHIPS BETWEEN CONTINUED FRACTION IDENTITIES, q-PRODUCT IDENTITIES AND COMBINATORIAL PARTITION IDENTITIES." Far East Journal of Mathematical Sciences (FJMS) 102, no. 4 (August 17, 2017): 855–63. http://dx.doi.org/10.17654/ms102040855.

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48

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, no. 2 (June 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 close connections with combinatorial partition-theoretic identities.

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49

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 (December 1, 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 known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

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50

Simsek, Yilmaz. "Combinatorial identities associated with Bernstein type basis functions." Filomat 30, no. 7 (2016): 1683–89. http://dx.doi.org/10.2298/fil1607683s.

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In this paper, we give some identities and relations for the Bernstein basis functions and the beta type polynomials. Integrating these identities, we derive many identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients and the Catalan numbers. We also give remarks and comments on these identities.

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