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Bertrand’s Ballot Theorem Cover

Bertrand’s Ballot Theorem

Formalized Mathematics
Volume 22 (2014): Issue 2 (June 2014)
By: Karol Pąk  
Open Access
|Jun 2014

References

  1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
    Search in Google Scholar
  2. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
    Search in Google Scholar
  3. [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
    Search in Google Scholar
  4. [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
    Search in Google Scholar
  5. [5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.
    Search in Google Scholar
  6. [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
    Search in Google Scholar
  7. [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
    Search in Google Scholar
  8. [8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.
    Search in Google Scholar
  9. [9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
    Search in Google Scholar
  10. [10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
    Search in Google Scholar
  11. [11] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181–187, 2005.
    Search in Google Scholar
  12. [12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.
    Search in Google Scholar
  13. [13] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.
    Search in Google Scholar
  14. [14] Karol Pąk. Cardinal numbers and finite sets. Formalized Mathematics, 13(3):399–406, 2005.
    Search in Google Scholar
  15. [15] Karol Pąk. The Catalan numbers. Part II. Formalized Mathematics, 14(4):153–159, 2006. doi:10.2478/v10037-006-0019-7.10.2478/v10037-006-0019-7
    Search in Google Scholar
  16. [16] Jan Popiołek. Introduction to probability. Formalized Mathematics, 1(4):755–760, 1990.
    Search in Google Scholar
  17. [17] M. Renault. Four proofs of the ballot theorem. Mathematics Magazine, 80(5):345–352, December 2007.10.1080/0025570X.2007.11953509
    Search in Google Scholar
  18. [18] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
    Search in Google Scholar
  19. [19] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.
    Search in Google Scholar
  20. [20] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5 (3):317–322, 1996.
    Search in Google Scholar
  21. [21] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.
    Search in Google Scholar
  22. [22] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
    Search in Google Scholar
  23. [23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
    Search in Google Scholar
  24. [24] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.
    Search in Google Scholar
  25. [25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
    Search in Google Scholar
  26. [26] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.
    Search in Google Scholar
DOI: https://doi.org/10.2478/forma-2014-0014 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
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Language: English
Page range: 119 - 123
Submitted on: Jun 13, 2014
|
Published on: Jun 30, 2014
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
ballot theorem,
probability
Related subjects:
Mathematics,
General mathematics,
Computer sciences,
Computer sciences, other

© 2014 Karol Pąk, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

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