The Relationship of Multiset, Stirling Number, Bell Number, and Catalan Number
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Keywords:
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Counting, Enumeration, Multiset, Catalan Numbers, Stirling Numbers
Abstract
Catalan numbers is not as famous as Fibonacci numbers, however this number has own its beauty and arts. Catalan numbers was discovered by Ming Antu in 1730, however, this numbers is credited to Eugene Catalan when he was studying parentheses in 1838. Catalan numbers mostly occurs in counting or enumeration problems. The Catalan numbers can be defined in more than one forms, and the most famous form is Cn = 1/n+1(2nn). In this study we will discuss the multiset construction and the relationship of the results of Multiset with Stirling, Bell, and Catalan numbers.
References
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Riordan, J. (2012). Introduction to Combinatorial Analysis. Courier Corporation
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Saračević, M., S. Adamović, and E. Biševac (2018). Application of Catalan Numbers and the Lattice Path Combinatorial Problem in Cryptography. Acta Polytechnica Hungarica, 15(7); 91-110
Saračević, M., S. Adamović, V. Miškovic, N. Maček, and M. Šarac (2019). A Novel Approach to Steganography Based on the Properties of Catalan Numbers and Dyck Words. Future Generation Computer Systems, 100; 186–197
Saračević, M., M. Hadžić, and E. Korićanin (2017). Generating Catalan-Keys Based on Dynamic Programming and their Application in Steganography. International Journal of Industrial Engineering and Management, 8(4); 219
Selim, A. and M. Saračević (2019). Catalan Numbers and Applications. International Scientific Journal Vision, 4(1); 99–114
Shapiro, L. W. (1976). A Catalan Triangle. Discrete Mathematics, 14(1); 83–90
Singmaster, D. (1978). An Elementary Evaluation of the Catalan Numbers. The American Mathematical Monthly, 85(5); 366–368
Stanley, R. P. (2015). Catalan Numbers. Cambridge University Press
Stojadinovic, T. (2015). On Catalan Numbers. The Teaching of Mathematics, 18(1); 16–24
Tripathy, B. C., S. Debnath, and D. Rakshit (2018). On Multiset Group. Proyecciones (Antofagasta), 37(3); 479–489
Bayer, M. and K. Brandt (2014). The Pill Problem, Lattice Paths, and Catalan Numbers. Mathematics Magazine, 87(5); 388–394
Boyadzhiev, K. N. (2021). Exotic Series with Bernoulli, harmonic, Catalan, and Stirling Numbers. arXiv Preprint arXiv:2110.00689; 1–16
Cayley, C. (1890). On the Partitions of a Polygon. Proceedings of the London Mathematical Society, 1(1); 237–264
Ceballos, C. and R. S. G. D’León (2018). Signature Catalan Combinatorics. arXiv Preprint arXiv:1805.03863
Guy, R. (1993). Parker’s Permutation Problem Involves the Catalan Numbers. The American Mathematical Monthly, 100(3); 287–289
İlkhan, M. (2020). A New Conservative Matrix Derived by Catalan Numbers and its Matrix Domain in the Spaces c and c0. Linear and Multilinear Algebra, 68(2); 417–434
Koshy, T. (2008). Catalan Numbers with Applications. Oxford University Press
Lee, K. W. and S. Oh (2018). Contemporary Mathematics, Vol. 712, pp. 165-185. American Mathematical Society
Ndagijimana, S. (2016). On Some Properties of Catalan Numbers and Application in RNA Secondary Structure. Ph.D. thesis, University of Rwanda
Pak, I. (2014). History of Catalan Numbers. arXiv Preprint arXiv:1408.5711
Riordan, J. (2012). Introduction to Combinatorial Analysis. Courier Corporation
Roman, S. (2005). Advanced Linier Algebra. Springer
Saračević, M., S. Adamović, and E. Biševac (2018). Application of Catalan Numbers and the Lattice Path Combinatorial Problem in Cryptography. Acta Polytechnica Hungarica, 15(7); 91-110
Saračević, M., S. Adamović, V. Miškovic, N. Maček, and M. Šarac (2019). A Novel Approach to Steganography Based on the Properties of Catalan Numbers and Dyck Words. Future Generation Computer Systems, 100; 186–197
Saračević, M., M. Hadžić, and E. Korićanin (2017). Generating Catalan-Keys Based on Dynamic Programming and their Application in Steganography. International Journal of Industrial Engineering and Management, 8(4); 219
Selim, A. and M. Saračević (2019). Catalan Numbers and Applications. International Scientific Journal Vision, 4(1); 99–114
Shapiro, L. W. (1976). A Catalan Triangle. Discrete Mathematics, 14(1); 83–90
Singmaster, D. (1978). An Elementary Evaluation of the Catalan Numbers. The American Mathematical Monthly, 85(5); 366–368
Stanley, R. P. (2015). Catalan Numbers. Cambridge University Press
Stojadinovic, T. (2015). On Catalan Numbers. The Teaching of Mathematics, 18(1); 16–24
Tripathy, B. C., S. Debnath, and D. Rakshit (2018). On Multiset Group. Proyecciones (Antofagasta), 37(3); 479–489
Authors
Wamiliana, Yuliana, A., & Fitriani. (2023). The Relationship of Multiset, Stirling Number, Bell Number, and Catalan Number. Science and Technology Indonesia, 8(2), 330–337. https://doi.org/10.26554/sti.2023.8.2.330-337
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