In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose so that to indicate easily by name their meaning. The states are separated into two classes: non-nominal states and nominal states. The states are related with the place I, II, III etc on arrival line. It is necessary to generate the total number of non-nominal states (on arrival line) and the total number of nominal states. In order to generate the states the work uses some formulas and some specialised algorithms. For example, the consrtuction of all non-nominal states recommends that the string for the position I to use a decreasing string. The same rule is validly for position II, but for sub-strings etc. A lot of numerical examples ilustrate the states generation. An independent method verifies the correctitude of states generation. In order to continue the study of combinatorial problem, the work introduces two new notions in section 5. The notions of partial frequency and final frequency are defined for a nominal known runner in final classification, together with computational formulas. The section 6 constructs the random variables attached to final classification and the probability of each place on arrival line. Each runner receives a score (a number of points) related with his final classification. May be the runner is interested to know the probability to ocuppy the first place (place I) and to estimate the number of possible points. All the results could be written in a centralisation table (section 7). Section 8 contains several numerical examples with statistical computations. At the end of the work we replace hypothesis 1 by hypothesis 2: only one runner could ocuppay each place. All the above notions have a new specific form. The numerical examples ilustrates the theory.
| Published in | Applied and Computational Mathematics (Volume 10, Issue 3) |
| DOI | 10.11648/j.acm.20211003.11 |
| Page(s) | 46-55 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Arrival Line, Non-nominal State, Nominal State, Partial Feequency, Final Frequency, Algorithm of Arrival Line, Combinatorial Problem, Statistical Problem
| [1] | Popoviciu Nicolae, (2017). A Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners; Proceedings of ENEC Confference, Editor in Chief Ion Spanulescu; May 2017, Hyperion University, Bucharest, Romania. |
| [2] | Popoviciu Nicolae, (2014). Capitole fundamentale de probabilități și statistică matematică; (Special Chapters of Probability and Statistics). Editura Victor, Universitatea Hyperion din București, ISBN 978-973-1815-98-5. |
| [3] | Bona, Miklos; (2005). A Walk Through Combinatorics: An Itroduction to Enumeration and Graph Theory; World Scientific Publishing; ISBN 981-02-49000-4. |
| [4] | Nathanson M, B; (2000). Elementary Methods in Number Theory; 195; Springer-Verlag; ISBN 0-387-98912-9. |
| [5] | Riordan, J.; (1958), An Introduction to Combinatorial Analysis, Wiley. |
| [6] | Hall, M., (1967), Combinatorial Theory, Blaisdell. |
| [7] | Beckenbach, E. F.(ed); (1964), Applied Combinatorial Mathematics, Wiley. |
| [8] | Erdos, P.; (1974), Probabilistic Methods in Combinatorics, Acad. Press. |
| [9] | Sachov, V. N.; (1977), Combinatorial Methods in Discrete Mathematics, Moscow (in Russian). |
| [10] | Sachov, V. N.; (1978), Probabilistic Methods in Combinatorial Analysiss, Moscow (in Russian). N Popoviciu Revised 24 May 2021. |
APA Style
Nicolae Popoviciu. (2021). Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners. Applied and Computational Mathematics, 10(3), 46-55. https://doi.org/10.11648/j.acm.20211003.11
ACS Style
Nicolae Popoviciu. Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners. Appl. Comput. Math. 2021, 10(3), 46-55. doi: 10.11648/j.acm.20211003.11
AMA Style
Nicolae Popoviciu. Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners. Appl Comput Math. 2021;10(3):46-55. doi: 10.11648/j.acm.20211003.11
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Download@article{10.11648/j.acm.20211003.11,
author = {Nicolae Popoviciu},
title = {Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners},
journal = {Applied and Computational Mathematics},
volume = {10},
number = {3},
pages = {46-55},
doi = {10.11648/j.acm.20211003.11},
url = {https://doi.org/10.11648/j.acm.20211003.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211003.11},
abstract = {In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose so that to indicate easily by name their meaning. The states are separated into two classes: non-nominal states and nominal states. The states are related with the place I, II, III etc on arrival line. It is necessary to generate the total number of non-nominal states (on arrival line) and the total number of nominal states. In order to generate the states the work uses some formulas and some specialised algorithms. For example, the consrtuction of all non-nominal states recommends that the string for the position I to use a decreasing string. The same rule is validly for position II, but for sub-strings etc. A lot of numerical examples ilustrate the states generation. An independent method verifies the correctitude of states generation. In order to continue the study of combinatorial problem, the work introduces two new notions in section 5. The notions of partial frequency and final frequency are defined for a nominal known runner in final classification, together with computational formulas. The section 6 constructs the random variables attached to final classification and the probability of each place on arrival line. Each runner receives a score (a number of points) related with his final classification. May be the runner is interested to know the probability to ocuppy the first place (place I) and to estimate the number of possible points. All the results could be written in a centralisation table (section 7). Section 8 contains several numerical examples with statistical computations. At the end of the work we replace hypothesis 1 by hypothesis 2: only one runner could ocuppay each place. All the above notions have a new specific form. The numerical examples ilustrates the theory.},
year = {2021}
}
TY - JOUR T1 - Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners AU - Nicolae Popoviciu Y1 - 2021/06/22 PY - 2021 N1 - https://doi.org/10.11648/j.acm.20211003.11 DO - 10.11648/j.acm.20211003.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 46 EP - 55 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20211003.11 AB - In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose so that to indicate easily by name their meaning. The states are separated into two classes: non-nominal states and nominal states. The states are related with the place I, II, III etc on arrival line. It is necessary to generate the total number of non-nominal states (on arrival line) and the total number of nominal states. In order to generate the states the work uses some formulas and some specialised algorithms. For example, the consrtuction of all non-nominal states recommends that the string for the position I to use a decreasing string. The same rule is validly for position II, but for sub-strings etc. A lot of numerical examples ilustrate the states generation. An independent method verifies the correctitude of states generation. In order to continue the study of combinatorial problem, the work introduces two new notions in section 5. The notions of partial frequency and final frequency are defined for a nominal known runner in final classification, together with computational formulas. The section 6 constructs the random variables attached to final classification and the probability of each place on arrival line. Each runner receives a score (a number of points) related with his final classification. May be the runner is interested to know the probability to ocuppy the first place (place I) and to estimate the number of possible points. All the results could be written in a centralisation table (section 7). Section 8 contains several numerical examples with statistical computations. At the end of the work we replace hypothesis 1 by hypothesis 2: only one runner could ocuppay each place. All the above notions have a new specific form. The numerical examples ilustrates the theory. VL - 10 IS - 3 ER -
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