SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM

Kubilay Dagtoros

doi:10.54559/jauist.1070936

Research Article

SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM

Year 2022, Volume: 3 Issue: 1, 1 - 7, 30.06.2022

Kubilay Dagtoros

https://doi.org/10.54559/jauist.1070936

Abstract

We present a recursive formula for the two-sided ballot theorem using left and right shift transforms. In particular, we showed that the xth entry of the image of the d + 1 dimensional unit vector under the sum of the left and right shift operators is the number of walks in the lattice interval [0,d] that start at the origin and stop at the location x. This approach enables us to write a recursive formula for the number of possible n−walks between two obstacles that stop at a predetermined location.

Keywords

ballot theorem, random walk, self-avoiding walk, reflection method.

References

[1] A. Aeppli, Zur Theorie verketteter Wahrscheinlichkeitem, Markoffsche Ketten h ̈oherer Ordnung, Ph.D. Thesis, Eidgenössische Technische Hochschule, Zürich, 1924.
[2] D. Andr ́e, Solution directe du probl`eme r ́esolu par M. Bertrand, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) 436-437.
[3] E ́. Barbier, G ́en ́eralisation du probl`eme r ́esolu par M. J. Bertrand, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) p.407.
[4] J. Bertrand, Solution d’un probl`eme, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) p.369.
[5] W. Feller, An introduction to probability theory and its applications, Vol. 1, (third edition, revised), John Wiley and Sons, 1970.
[6] P. A. MacMahon, Memoir on the theory of the partitions of numbers, part iv: on the probability that the successful candidate at an election by ballot may never at anytime have fewer votes than the one who is unsuccessful; on a generalization of this question; and its connection with other questions of partition, permutation, and combination, Philosophical Transactions of the Royal Society of London, Series A 209 (1909) 153- 175.
[7] T. V. Narayana, Lattice path combinatorics with statistical applications, University of Toronto Press, 1979.
[8] M. Renault, Four proofs of the ballot theorem, Math. Mag. 80 (2007), 345-352.
[9] R. Srinivasan, On some results of Taka ́cs in ballot problems, Discrete Math. 28 (1979), 213-218.
[10] L. Tak ́acs, On the ballot theorems, Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkha ̈user, 1997.

Year 2022, Volume: 3 Issue: 1, 1 - 7, 30.06.2022

Kubilay Dagtoros

https://doi.org/10.54559/jauist.1070936

Abstract

References

[1] A. Aeppli, Zur Theorie verketteter Wahrscheinlichkeitem, Markoffsche Ketten h ̈oherer Ordnung, Ph.D. Thesis, Eidgenössische Technische Hochschule, Zürich, 1924.
[2] D. Andr ́e, Solution directe du probl`eme r ́esolu par M. Bertrand, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) 436-437.
[3] E ́. Barbier, G ́en ́eralisation du probl`eme r ́esolu par M. J. Bertrand, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) p.407.
[4] J. Bertrand, Solution d’un probl`eme, Comptes Rendus de l’Acad ́emie des Sciences, Paris 105 (1887) p.369.
[5] W. Feller, An introduction to probability theory and its applications, Vol. 1, (third edition, revised), John Wiley and Sons, 1970.
[6] P. A. MacMahon, Memoir on the theory of the partitions of numbers, part iv: on the probability that the successful candidate at an election by ballot may never at anytime have fewer votes than the one who is unsuccessful; on a generalization of this question; and its connection with other questions of partition, permutation, and combination, Philosophical Transactions of the Royal Society of London, Series A 209 (1909) 153- 175.
[7] T. V. Narayana, Lattice path combinatorics with statistical applications, University of Toronto Press, 1979.
[8] M. Renault, Four proofs of the ballot theorem, Math. Mag. 80 (2007), 345-352.
[9] R. Srinivasan, On some results of Taka ́cs in ballot problems, Discrete Math. 28 (1979), 213-218.
[10] L. Tak ́acs, On the ballot theorems, Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkha ̈user, 1997.

There are 10 citations in total.

Details

Primary Language	English
Subjects	Engineering
Journal Section	Research & Review Articles
Authors	Kubilay Dagtoros
Publication Date	June 30, 2022
Published in Issue	Year 2022 Volume: 3 Issue: 1

Cite

APA	Dagtoros, K. (2022). SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM. Journal of Amasya University the Institute of Sciences and Technology, 3(1), 1-7. https://doi.org/10.54559/jauist.1070936
AMA	Dagtoros K. SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM. J. Amasya Univ. Inst. Sci. Technol. June 2022;3(1):1-7. doi:10.54559/jauist.1070936
Chicago	Dagtoros, Kubilay. “SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM”. Journal of Amasya University the Institute of Sciences and Technology 3, no. 1 (June 2022): 1-7. https://doi.org/10.54559/jauist.1070936.
EndNote	Dagtoros K (June 1, 2022) SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM. Journal of Amasya University the Institute of Sciences and Technology 3 1 1–7.
IEEE	K. Dagtoros, “SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM”, J. Amasya Univ. Inst. Sci. Technol., vol. 3, no. 1, pp. 1–7, 2022, doi: 10.54559/jauist.1070936.
ISNAD	Dagtoros, Kubilay. “SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM”. Journal of Amasya University the Institute of Sciences and Technology 3/1 (June 2022), 1-7. https://doi.org/10.54559/jauist.1070936.
JAMA	Dagtoros K. SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM. J. Amasya Univ. Inst. Sci. Technol. 2022;3:1–7.
MLA	Dagtoros, Kubilay. “SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM”. Journal of Amasya University the Institute of Sciences and Technology, vol. 3, no. 1, 2022, pp. 1-7, doi:10.54559/jauist.1070936.
Vancouver	Dagtoros K. SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM. J. Amasya Univ. Inst. Sci. Technol. 2022;3(1):1-7.