Combinatorial Properties of Two Families of Generalized Eule
發(fā)布時(shí)間:2021-04-09 06:27
歐拉多項(xiàng)式是組合數(shù)學(xué)中一個(gè)極其重要的多項(xiàng)式,并且歐拉數(shù)也是一項(xiàng)被廣為研究的序列,它們?cè)诮M合數(shù)學(xué)中都有著非常重要的應(yīng)用和意義,而關(guān)于歐拉多項(xiàng)式的推廣是近年來一個(gè)重要的研究問題.Brenti和Welker給出了限制歐拉多項(xiàng)式和限制歐拉數(shù)的相關(guān)定義和性質(zhì),它們是在經(jīng)典歐拉多項(xiàng)式和歐拉數(shù)的基礎(chǔ)上通過限制排列的第一位而得到的,限制歐拉多項(xiàng)式又被Petersen稱為j-歐拉多項(xiàng)式.此外,Nunge還定義了分段排列上的廣義歐拉多項(xiàng)式和歐拉數(shù),并進(jìn)一步提出了這類廣義歐拉多項(xiàng)式系數(shù)的單峰性猜想.本文在對(duì)兩類廣義歐拉多項(xiàng)式展開深入研究后,利用Bor cea與B randen所推廣的多元?dú)W拉多項(xiàng)式的穩(wěn)定性理論證明了 Nunge在分段排列上定義的廣義歐拉多項(xiàng)式的單峰性猜想.此外,還基于Brenti和Welker所定義的限制歐拉多項(xiàng)式的已有性質(zhì),提出了這類廣義歐拉多項(xiàng)式的幾種組合解釋,并進(jìn)一步定義了一類新的限制歐拉多項(xiàng)式,對(duì)Brenti所提出的限制歐拉多項(xiàng)式作了進(jìn)一步推廣.具體結(jié)構(gòu)如下:第一章簡要介紹了研究背景,對(duì)歐拉多項(xiàng)式,分段排列上的廣義歐拉多項(xiàng)式,限制歐拉多項(xiàng)式,多元多項(xiàng)式的穩(wěn)定性理論以及Carlitz等...
【文章來源】:天津師范大學(xué)天津市
【文章頁數(shù)】:44 頁
【學(xué)位級(jí)別】:碩士
【文章目錄】:
摘要
Abstract
Chapter 1 Introduction
1.1 Eulerian polynomials
1.2 Generalized Eulerian polynomials on segmented permutations
1.3 Restricted Eulerian polynomials
1.4 Structure of the paper
Chapter 2 Generalized Eulerian polynomials on segmented permutations
2.1 Multivariate stable polynomials
n(t),Kn,j(x) and Ln,i(x)"> 2.2 Stability of Pn(t),Kn,j(x) and Ln,i(x)
n(t),Kn,j(x) and Ln,i(x)"> 2.3 Recurrence relations of Pn(t),Kn,j(x) and Ln,i(x)
Chapter 3 Restricted Eulerian polynomials
3.1 (i,j)-Eulcrian polynomials
n,j"> 3.2 The equidistribution of excedances and descents in the group of An,j
3.3 The generalization of Carlitz identity
Conclusion
Reference
致謝
攻讀學(xué)位期間發(fā)表的學(xué)術(shù)論文
本文編號(hào):3127100
【文章來源】:天津師范大學(xué)天津市
【文章頁數(shù)】:44 頁
【學(xué)位級(jí)別】:碩士
【文章目錄】:
摘要
Abstract
Chapter 1 Introduction
1.1 Eulerian polynomials
1.2 Generalized Eulerian polynomials on segmented permutations
1.3 Restricted Eulerian polynomials
1.4 Structure of the paper
Chapter 2 Generalized Eulerian polynomials on segmented permutations
2.1 Multivariate stable polynomials
n(t),Kn,j(x) and Ln,i(x)"> 2.2 Stability of Pn(t),Kn,j(x) and Ln,i(x)
n(t),Kn,j(x) and Ln,i(x)"> 2.3 Recurrence relations of Pn(t),Kn,j(x) and Ln,i(x)
Chapter 3 Restricted Eulerian polynomials
3.1 (i,j)-Eulcrian polynomials
n,j"> 3.2 The equidistribution of excedances and descents in the group of An,j
Conclusion
Reference
致謝
攻讀學(xué)位期間發(fā)表的學(xué)術(shù)論文
本文編號(hào):3127100
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