臺灣博碩士論文加值系統

在本論文中，我們研究了限制高度之施羅德路徑的漢克爾行列式。$s^{(k)}_n$表示限制高度k且長度為n的施羅德路徑數，其中施羅德路徑是一個位於第一象限的格線路徑，以原點(0,0)為起點，終點為(2n,0)，可使用的單位步有上升步U=(1,1)，下降步D=(1,-1)以及水平步H=(2,0)。限制高度k之施羅德路徑永遠不會超過y=0以下，且不超過y=k以上。給定任意序列$A={a_n}$，$H^{(l)}_n$代表的是n×n的漢克爾矩陣，則$H_n^{(l)}(A) = {({a_{i + j + l}})_{0 le i,j le n - 1}}$。

對於漢克爾行列式已有很多的研究如[5][6][10][11]。在本篇論文中，我們將計算其有限制高度之施羅德路徑的漢克爾行列式。

在第二章中，我們介紹有名的格線路徑如Dyck path，Motzkin path和施羅德路徑的漢克爾行列式。

在第三章中，我們將介紹有限之高度之施羅德路徑的生成函數及與Delannoy numbers 的關係。

在第四章中，我們將給予一個限制高度之施羅德路徑的漢克爾行列式的值。

In this thesis we study the Hankel determinants for Schröder paths with restricted height. Let $s^{(k)}_n$
n denote the number of Schröder paths with restricted height k, which those lattice paths that are in the first quadrant, begin at the (0, 0), end on the (2n, 0), consist of up steps U = (1, 1), down steps D = (1,−1), and level steps H = (2, 0). The Schröder paths with restricted height k which never run below the horizontal path y = 0, and never run over the horizontal path y = k. For a given sequence $A = {a_n}$, let $H^{(l)}_n$ denote the n by n Hankel matrix, defined that $H_n^{(l)}(A) = {({a_{i + j + l}})_{0 le i,j le n - 1}}$.

The Hankel determinants has received a lot of attention [5][6][10][11]. In this thesis, we evaluate Hankel determinants for Schröder path with restricted height.

In chapter 2, we will introduce the known results of the Hankel determinants for Dyck path, Motzkin path and Schröder path.

In chapter 3, we will show the generating function of the Schröder paths with restricted height, and the relation between Delannoy numbers and Schröder numbers.

In chapter 4, we will show the determinants values for Hankel matrix of the Schröder path with restricted height.

[論文審定書+i]
[Chinese Abstract+iii]
[Abstract+iv]
[Contents+v]
[List of Figures+vii]
[1 Introduction+1]
[1.1 The basic notations+1]
[1.2 Themain result of the thesis+3]
[2 The Known Results+5]
[2.1 Dyck path+5]
[2.2 Motzkin path+7]
[2.3 Schr¨oder path+8]
[2.4 The key lemma+10]
[2.4.1 Hankel determinants for Dyck path+10]
[2.4.2 Hankel determinants for Motzkin path+12]
[2.4.3 Hankel determinants for Schröder path+12]
[3 The generating function of Schröder path with restricted
height+14]
[3.1 Schröder path with restricted height+14]
[3.2 Proof of the theorem1.1+15]
[3.3 Delannoy triangle+16]
[4 Hankel determinants for Schröder path with restricted
height+18]
[4.1 The proof of theorem1.2+18]
[Reference+24]

Aigner, M. (2007). A course in enumeration (Vol. 238). Springer Science & Business Media.
Aigner, M. (1999). Catalan-like numbers and determinants. Journal of Combinatorial Theory, Series A, 87(1), 33-51.
Bressoud, D. M. (1999). Proofs and Con firmations: The Story of the Alternating-Sign Matrix Conjecture. Cambridge University Press.
Brualdi, R. A., Kirkland, S. (2005). Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers. Journal of Combinatorial Theory, Series B, 94(2), 334-351.
Cameron, N. T., Yip, A. C. (2011). Hankel determinants of sums of consecutive Motzkin numbers. Linear algebra and its applications, 434(3), 712-722.
Eu, S. P., Wong, T. L., Yen, P. L. (2012). Hankel determinants of sums of consecutive weighted Schröder numbers. Linear Algebra and its Applications, 437(9), 2285-2299.
Gessel, I., Viennot, G. (1985). Binomial determinants, paths, and hook length formulae. Advances in mathematics, 58(3), 300-321.
Gessel, I. M., Viennot, X. (1989). Determinants, paths, and plane partitions. preprint, 132(197.15).
Gessel, I., Xin, G. (2006). The generating function of ternary trees and continued fractions. Electron. J. Combin, 13(1), R53.
Sulanke, R. A., Xin, G. (2008). Hankel determinants for some common lattice paths. Advances in Applied Mathematics, 40(2), 149-167.
Woan, W. J. (2001). Hankel matrices and lattice paths. J. Integer Seq, 4, Article 01.1.2.