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題名 Some Results on Path Pairs 作者 劉洪鈞 貢獻者 李陽明
劉洪鈞關鍵詞 Path Pairs ; Non - intersecting Paths; 日期 2002 上傳時間 11-Oct-2018 11:50:48 (UTC+8) 摘要 In this thesis, our goal is to use mathematical induction to give a direct proof to show that the number of b(n - m, k ; n, k - m) is m/(n+k-m) ,where b(n – m, k; n, k - m) denotes the number of non-intersecting paths that the upper path goes from (0, 0) to(n - m, k) while the lower path goes from (0, 0) to (n, k - m). Furthermore, we conclude two applications about b(n-m, k ; n, k-m), namely b(n, k) (see Definition 2.2) and PP(n, k) (see Definition 4.4). We also bring up some open problems concerning our topics. 參考文獻 References [1] Bessenrodt, C., “On hooks of Young diagrams”, Annals of Combinatorics 2 (1998), 103-110. [2] Franzblau, D. and Zeilberger, D., “A bijective proof of the hook-length formula”, Journal of Algorithms 3 (1982), 317-342. [3] Goulden, I. P. and Jackson, D. M., Combinatorial Enumeration, John Wiley & Sons, 1983. [4] Greene, C., Nijenhuis, A. and Wilf, H. S., “A probabilistic proof of a formula for the number of Young tableaux of a given shape”, Adv. in Math 31 (1979),104-109. [5] Grimaldi, Ralph P., Discrete and Combinatorial Mathematics: An Applied Introduction, 3nd ed., Addison-Wesley, 1994. [6] Hillman, A. P. and Grassl, R. M., “Reverse plane partition and tableau hook numbers”, Journal of Combinatorial Theory 21 (1976), 216-221. [7] Knuth, Donald E., “Permutations, matrices and generalized Young tableaux”, Pac. J. Math 34 (1970). [8] Knuth, Donald E., The Art Of Computer Programming, Vol. 3, Sorting and Searching, 2nd ed., Addison-Wesley, 1997. [9] Krattenthaler, C., “The major counting of nonintersecting lattice paths and generating functions for tableaux”, Memoirs of the American Mathematical Society (1995), Vol 115, Number 552. [10] Levine, J., “Note on the number of pairs of non-intersecting routes”, Scripta Mathematica 24 (1959), 335-338. [11] Liu, C. L., Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. [12] Narayana, T. V., Lattice path combinatorics with statistical applications, University of Toronto Press, 1979. [13] Nijenhuis, A. and Wilf, H. S., Combinatorial Algorithms, 2nd ed., Academic Press, New York, 1978. [14] Pólya, G., “On the number of certain lattice polygons”, Journal of Combinatorial Theory6 (1969), 102-105. [15] Regev, A. and Zeilberger, D., “Proof of a Conjecture on Multisets of Hook Numbers”, Annals of Combinatorics 1 (1997), 391-394. [16] Riordan, J., Combinatorial Identities, John Wiley & Sons, 1968. [17] Shapiro, L. W., “A Catalan triangle”, Discrete Mathematics 14 (1976), 83-90. [18] William, F., Young Tableaux: with applications to representation theory and geometry, Cambridge University Press, New York, 1997. [19] Woan, W. J., Shapiro, L., Rogers, D. G., “The Catalan numbers, the Lebesgue integral, and 4n-2”, Am. Math. Monthly 104 (1997), 10. [20] Woan, W. J., “Area of Catalan paths”, Discrete Mathematics 226 (2001),439-444. [21] Zeilberger, D., “A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof”, Discrete Mathematics 51 (1984), 101-108. 描述 碩士
國立政治大學
應用數學系
90資料來源 http://thesis.lib.nccu.edu.tw/record/#G91NCCV3412012 資料類型 thesis dc.contributor.advisor 李陽明 - dc.contributor.author (Authors) 劉洪鈞 - dc.creator (作者) 劉洪鈞 - dc.date (日期) 2002 - dc.date.accessioned 11-Oct-2018 11:50:48 (UTC+8) - dc.date.available 11-Oct-2018 11:50:48 (UTC+8) - dc.date.issued (上傳時間) 11-Oct-2018 11:50:48 (UTC+8) - dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/120511 - dc.description (描述) 碩士 - dc.description (描述) 國立政治大學 - dc.description (描述) 應用數學系 - dc.description (描述) 90 - dc.description.abstract (摘要) In this thesis, our goal is to use mathematical induction to give a direct proof to show that the number of b(n - m, k ; n, k - m) is m/(n+k-m) ,where b(n – m, k; n, k - m) denotes the number of non-intersecting paths that the upper path goes from (0, 0) to(n - m, k) while the lower path goes from (0, 0) to (n, k - m). Furthermore, we conclude two applications about b(n-m, k ; n, k-m), namely b(n, k) (see Definition 2.2) and PP(n, k) (see Definition 4.4). We also bring up some open problems concerning our topics. - dc.description.tableofcontents Abstract i 1 Introduction 1 1.1 Importance of this study 1 1.2 Purpose of this study 1 1.3 Structure of this study 1 2 Literature review 3 3 The Number of b(n - m, k;n, k - m) 7 3.1 The recurrence relation of b(n - m, k;n, k - m) 7 3.2 The proof of b(n - m, k;n, k - m) 9 3.3 Example 4 Applications of b(n - m, k ; n, k - m) 14 4.1 The number of b(n, k) 14 4.2 The number of PP(n, k) 14 5 Conclusion 26 References 29 - dc.format.extent 115 bytes - dc.format.mimetype text/html - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G91NCCV3412012 - dc.subject (關鍵詞) Path Pairs ; Non - intersecting Paths; - dc.title (題名) Some Results on Path Pairs - dc.type (資料類型) thesis - dc.relation.reference (參考文獻) References [1] Bessenrodt, C., “On hooks of Young diagrams”, Annals of Combinatorics 2 (1998), 103-110. [2] Franzblau, D. and Zeilberger, D., “A bijective proof of the hook-length formula”, Journal of Algorithms 3 (1982), 317-342. [3] Goulden, I. P. and Jackson, D. M., Combinatorial Enumeration, John Wiley & Sons, 1983. [4] Greene, C., Nijenhuis, A. and Wilf, H. S., “A probabilistic proof of a formula for the number of Young tableaux of a given shape”, Adv. in Math 31 (1979),104-109. [5] Grimaldi, Ralph P., Discrete and Combinatorial Mathematics: An Applied Introduction, 3nd ed., Addison-Wesley, 1994. [6] Hillman, A. P. and Grassl, R. M., “Reverse plane partition and tableau hook numbers”, Journal of Combinatorial Theory 21 (1976), 216-221. [7] Knuth, Donald E., “Permutations, matrices and generalized Young tableaux”, Pac. J. Math 34 (1970). [8] Knuth, Donald E., The Art Of Computer Programming, Vol. 3, Sorting and Searching, 2nd ed., Addison-Wesley, 1997. [9] Krattenthaler, C., “The major counting of nonintersecting lattice paths and generating functions for tableaux”, Memoirs of the American Mathematical Society (1995), Vol 115, Number 552. [10] Levine, J., “Note on the number of pairs of non-intersecting routes”, Scripta Mathematica 24 (1959), 335-338. [11] Liu, C. L., Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. [12] Narayana, T. V., Lattice path combinatorics with statistical applications, University of Toronto Press, 1979. [13] Nijenhuis, A. and Wilf, H. S., Combinatorial Algorithms, 2nd ed., Academic Press, New York, 1978. [14] Pólya, G., “On the number of certain lattice polygons”, Journal of Combinatorial Theory6 (1969), 102-105. [15] Regev, A. and Zeilberger, D., “Proof of a Conjecture on Multisets of Hook Numbers”, Annals of Combinatorics 1 (1997), 391-394. [16] Riordan, J., Combinatorial Identities, John Wiley & Sons, 1968. [17] Shapiro, L. W., “A Catalan triangle”, Discrete Mathematics 14 (1976), 83-90. [18] William, F., Young Tableaux: with applications to representation theory and geometry, Cambridge University Press, New York, 1997. [19] Woan, W. J., Shapiro, L., Rogers, D. G., “The Catalan numbers, the Lebesgue integral, and 4n-2”, Am. Math. Monthly 104 (1997), 10. [20] Woan, W. J., “Area of Catalan paths”, Discrete Mathematics 226 (2001),439-444. [21] Zeilberger, D., “A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof”, Discrete Mathematics 51 (1984), 101-108. -
