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 Title: Some Results on Path Pairs Authors: 劉洪鈞 Contributors: 李陽明劉洪鈞 Keywords: Path Pairs;Non - intersecting Paths Date: 2002 Issue Date: 2018-10-11 11:50:48 (UTC+8) Abstract: 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. Reference: 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. Description: 碩士國立政治大學應用數學系90 Source URI: http://thesis.lib.nccu.edu.tw/record/#G91NCCV3412012 Data Type: thesis Appears in Collections: [應用數學系] 學位論文

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