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A Tale of Catalan Triangles: Counting Lattice Paths

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Soft Computing Applications (SOFA 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1438))

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Abstract

After we briefly review several different Catalan triangles, we use a particular type of lattice paths, namely brick-wall lattice paths, to construct a new triangular arrangement of integers containing the well-known sequence of Catalan numbers.

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References

  1. André, D.: Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris 105, 436–437 (1887)

    Google Scholar 

  2. Arbogast, L.F.A.: Du calcul des derivations (1800). https://archive.org/details/ducalculdesdriv00arbogoog/page/n245/mode/1up

  3. Bailey, D.F.: Counting arrangements of 1’s and -1’s. Math. Mag. 69(2), 128–131 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertrand, J.: Solution d’un problème. C. R. Acad. Sci. Paris 105, 369 (1887)

    MATH  Google Scholar 

  5. Catalan, E.: Note sur une équation aux différences finies. J. Math. Pures Appl. 3, 508–516 (1838)

    Google Scholar 

  6. Cowell, S., Beiu, V., Dăuş, L., Poulin, P.: On the exact reliability enhancements of small hammock networks. IEEE Access 6, 25411–25426 (2018)

    Article  Google Scholar 

  7. Cristescu, G., Drăgoi, V.: Efficient approximation of two-terminal networks reliability polynomials using cubic splines. IEEE Trans. Reliab. 70(3), 1193–1203 (2021)

    Article  Google Scholar 

  8. Cristescu, G., Drăgoi, V., Hoară, S.H.: Generalized convexity properties and shape-based approximation in networks reliability. Mathematics 9(24), 3182 (2021)

    Article  Google Scholar 

  9. Dăuş, L., Beiu, V., Cowell, S., Poulin, P.: Brick-wall lattice paths and applications. Technical report, arXiv (2018). http://arxiv.org/abs/1804.05277

  10. Dăuş, L., Jianu, M.: Full Hermite interpolation of the reliability of a hammock network. Appl. Anal. Discrete Math. 14, 198–220 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dăuş, L., Jianu, M.: The shape of the reliability polynomial of a hammock network. In: Dzitac, I., Dzitac, S., Filip, F.G., Kacprzyk, J., Manolescu, M.-J., Oros, H. (eds.) ICCCC 2020. AISC, vol. 1243, pp. 93–105. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-53651-0_8

    Chapter  Google Scholar 

  12. Drăgoi, V., Beiu, V.: Fast reliability ranking of matchstick minimal networks. Networks 79(4), 479–500 (2022)

    Article  MathSciNet  Google Scholar 

  13. Jianu, M., Ciuiu, D., Dăuş, L., Jianu, M.: Markov chain method for computing the reliability of hammock networks. Probab. Eng. Inf. Sci. 36(2), 276–293 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, J.: Ming Antu: the first inventor of Catalan numbers in the world. Acta Scientiarum Naturalium Universitatis Intramonglic 19(2), 239–245 (1988)

    Google Scholar 

  15. Luo, J.: Ming Antu and his power series expansions. In: Knobloch, E., Komatsu, H., Liu, D. (eds.) Seki, Founder of Modern Mathematics in Japan. Springer Proceedings in Mathematics & Statistics, vol. 39, pp. 299–310. Springer, Tokyo (2013). https://doi.org/10.1007/978-4-431-54273-5_20

    Chapter  Google Scholar 

  16. Koshy, T.: Catalan Numbers with Applications. Oxford University Press, New York (2009)

    MATH  Google Scholar 

  17. Miana, P.J., Romero, N.: Moments of combinatorial and Catalan numbers. J. Number Theory 130, 1876–1887 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays - Part I. J. Frankl. Inst. 262(3), 191–208 (1956)

    Article  MATH  Google Scholar 

  19. Pak, I.: History of Catalan numbers. Technical report, arXiv (2014). http://arxiv.org/abs/1408.5711v2

  20. Riordan, J.: Combinatorial Identities. John Wiley, New York (1968)

    MATH  Google Scholar 

  21. Ruskey, F., Hu, T.C.: Generating binary trees lexicographically. SIAM J. Comput. 6, 745–758 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shapiro, L.W.: A Catalan triangle. Discrete Math. 14, 83–90 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (OEIS). https://oeis.org

  24. Stanley, R.: Catalan Numbers. Cambridge University Press, New York (2015)

    Book  MATH  Google Scholar 

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Acknowledgments

This research was partly supported by the EU through the European Research Development Fund under the Competitiveness Operational Program (BioCell-NanoART = Novel Bio-inspired Cellular Nano-ARchiTectures, POC-A1-A1.1.4-E-2015 nr. 30/01.09.2016), and partly by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2495, within PNCDI III (ThUNDER2 = Techniques for Unconventional Nano-Designing in the Energy-Reliability Realm).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Technical University of Civil Engineering of Bucharest, 020396, Bucharest, Romania

    Leonard Dăuş & Marilena Jianu

  2. “Aurel Vlaicu” University of Arad, 310032, Arad, Romania

    Roxana-Mariana Beiu & Valeriu Beiu

Authors
  1. Leonard Dăuş
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  2. Marilena Jianu
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  3. Roxana-Mariana Beiu
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  4. Valeriu Beiu
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Corresponding author

Correspondence to Leonard Dăuş .

Editor information

Editors and Affiliations

  1. Faculty of Engineering, Department of Automatics and Applied Software, Aurel Vlaicu University of Arad, Arad, Romania

    Valentina Emilia Balas

  2. Aurel Vlaicu University of Arad, Arad, Romania

    Lakhmi C. Jain

  3. Faculty of Engineering, Department of Automatics and Applied Software, Aurel Vlaicu University of Arad, Arad, Romania

    Marius Mircea Balas

  4. Cankaya University, Ankara, Türkiye

    Dumitru Baleanu

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Dăuş, L., Jianu, M., Beiu, RM., Beiu, V. (2023). A Tale of Catalan Triangles: Counting Lattice Paths. In: Balas, V.E., Jain, L.C., Balas, M.M., Baleanu, D. (eds) Soft Computing Applications. SOFA 2020. Advances in Intelligent Systems and Computing, vol 1438. Springer, Cham. https://doi.org/10.1007/978-3-031-23636-5_52

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