本文選題：Chung-Feller定理 + 格路�。� 參考：《華東師范大學(xué)》2017年碩士論文

【摘要】：Irving和Rattan給出了在循環(huán)平移分段線(xiàn)性邊界控制下的格路個(gè)數(shù)的計(jì)算公式.他們的主要結(jié)論可以看作以下著名定理的一個(gè)推廣:從點(diǎn)(0,0)到點(diǎn)(kn,n)且被直線(xiàn)x=ky控制的格路個(gè)數(shù)(當(dāng)= 1時(shí),就是Dyck路).另一方面,著名的Chung-Feller定理告訴我們,從(0,0)到(n,n)且恰有2k(k= 0,1,…,n)個(gè)步法在直線(xiàn)y=x上方的格路個(gè)數(shù)與k的選取無(wú)關(guān),因此這樣的格路個(gè)數(shù)為Catalan數(shù)Cn=1/n+1(?).本篇論文主要研究在格路邊界對(duì)(P,a)下恰有k個(gè)瑕疵的格路個(gè)數(shù).其中P是從點(diǎn)(0,0)到點(diǎn)(n,m)的一條格路,a表示n的一個(gè)弱m-分拆,若格路P有k個(gè)向右的步法在邊界(?)a的上方,則稱(chēng)格路邊界對(duì)(P,a)有k個(gè)瑕疵.我們通過(guò)構(gòu)造一個(gè)雙射證明了,對(duì)于任意一個(gè)給定的分拆a,將所有在循環(huán)平移分段線(xiàn)性邊界(?)a控制下,滿(mǎn)足上述條件的格路加起來(lái)恰好是(?).也就是說(shuō),我們把Ivring-Rattan公式推廣到一個(gè)Chung-Feller類(lèi)型的定理.我們還將此雙射用于計(jì)算格路的雙上升數(shù)目,從而得到更為細(xì)致的結(jié)果.
[Abstract]:Irving and Rattan give the calculation formula of the number of lattice paths under the control of the circular translation piecewise linear boundary. Their main conclusions can be regarded as a generalization of the following famous theorems: the number of lattice paths controlled by the point (0,0) to point (KN, n) and by the linear x=ky (when = 1, it is Dyck Road). On the other hand, the famous Chung-Feller theorem tells me People, from (0,0) to (n, n) and there are just 2K (k= 0,1,... N) the number of lattice paths above the line y=x is independent of the selection of K, so the number of K is Catalan Cn=1/n+1 (?). This paper mainly deals with the number of K defects in the P, a, where P is a lattice from a point (0,0) to point (n, M). The step method is above the boundary (?) a, then it is called the lattice path boundary pair (P, a) with K defects. By constructing a double fire proof, for any given partition a, we add all the lattice paths that satisfy the above conditions under the cyclic translational piecewise linear boundary (?) a control. In other words, we extend the Ivring-Rattan formula. To a theorem of Chung-Feller type, we also use this double shot to calculate the number of double rise of grid path and get more detailed results.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O157

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 LI ShanHai;MA Jun;YEH YeongNan;;Uniform partition extensions,a generating functions perspective[J];Science China(Mathematics);2015年12期

2 趙天玉;楊方;;與Catalan數(shù)有關(guān)的組合問(wèn)題研究[J];長(zhǎng)江大學(xué)學(xué)報(bào)(自然科學(xué)版)理工卷;2008年04期

相關(guān)碩士學(xué)位論文 前1條

1 孫學(xué)芝;兩類(lèi)廣義Dyck路上的一些計(jì)數(shù)問(wèn)題[D];華東師范大學(xué);2014年

，

本文編號(hào)：1906151