Title Page
Contents
Abstract 11
Chapter 1. Introduction 13
1.1. Alternating permutations and q-Euler numbers 13
1.2. The poset of alternating permutations 15
1.3. The weak order on Coxeter groups 16
1.3.1. Structure of dissertation 17
Chapter 2. Preliminaries 19
2.1. Permutations 19
2.2. Posets 23
2.3. (P,ω)-partitions 25
2.4. Semistandard Young tableaux and reverse plane partitions 27
2.5. Coxeter groups and the weak order 28
Chapter 3. The q-Euler numbers and related combinatorial objects 33
3.1. Motivation and known results 33
3.2. Relations of the q-Euler numbers and other combinatorial objects 37
3.3. Bijective proofs of the identities involving the q-Euler numbers 40
3.3.1. Foata map 40
3.3.2. Modified version of the Foata map 42
3.3.3. Proofs of the identities 44
Chapter 4. The poset of alternating permutations ordered by the weak order 59
4.1. Poset structures 59
4.2. Rank generating function and the q-Euler number 62
4.3. The characteristic polynomial of the poset of alternating permutations 64
Chapter 5. Poset structures and characteristic polynomials of the weak order on Coxeter groups 69
5.1. The characteristic polynomial of an interval of a Coxeter group 69
5.2. Properties of descent classes 74
5.3. Permutations with a fixed descent set 77
Chapter 6. Modified characteristic polynomials for irreducible Coxeter groups 81
6.1. Modified characteristic polynomials for classical Coxeter groups 81
6.2. Modified characteristic polynomials for affine Coxeter groups 89
6.3. The modified characteristic polynomial for An[이미지참조] 101
References 107
논문요약 111
Table 3.1. Interpretations for Prodinger's q-tangent numbers. The notation Alt2n+1* means it can be either Alt2n+1 or Ralt2n+1.[이미지참조] 39
Table 3.2. Interpretations for Prodinger's q-secant numbers. The notation π* means it can be either πo or πe.[이미지참조] 40
Table 3.3. Each row defines the modified Foata map Fmod described in Theorem 3.3.1. If (C,D,≺)=(-,-,-), Fmod is the original Foata map.[이미지참조] 43
Figure 2.1. The Boolean poset B₃ and its dual. 24
Figure 2.2. The left of each element is a labeling ω and the right is a (P,ω)-partition. 26
Figure 2.3. The Young diagram of (5,4,2) on the left and the skew shape (5,4,2)/(2,1) on the right. 28
Figure 2.4. A semistandard Young tableau on the left, a reverse plane partition in the middle and a strict tableau on the right of shape (5,4,2)/(2,1). 28
Figure 3.1. The labelings ωλ/µSSYT on the left, ωλ/µRPP in the middle and ωλ/µST on the right for λ/µ=(5,4,2)/(2,1).[이미지참조] 34
Figure 3.2. The relation between the labels of three squares x,y and z for (Pδn+2/δn,ωδ n+2SSYT/δn).[이미지참조] 35
Figure 3.3. The connections in Theorems 3.2.2 and 3.2.3. 39
Figure 4.1. The poset Alt6 of alternating permutations.[이미지참조] 61
Figure 5.1. The Coxeter graph of An.[이미지참조] 78
Figure 6.1. The Coxeter graphs of Bn and Dn.[이미지참조] 85
Figure 6.2. The Coxeter graph of An containing the Coxeter graph of An.[이미지참조] 90
Figure 6.3. The Coxeter graph of An and the subset Q.[이미지참조] 91
Figure 6.4. The Coxeter graph of Bn containing the Coxeter graph of Bn.[이미지참조] 92
Figure 6.5. The Coxeter graph of Cn containing the Coxeter graph of Bn.[이미지참조] 93
Figure 6.6. The Coxeter graph of Dn containing the Coxeter graph of Dn.[이미지참조] 93
Figure 6.7. The Coxeter graph of E6 containing the Coxeter graph of E6.[이미지참조] 94
Figure 6.8. The Coxeter graph of E7 containing the Coxeter graph of E7.[이미지참조] 96
Figure 6.9. The Coxeter graph of E8 containing the Coxeter graph of E8.[이미지참조] 97
Figure 6.10. The Coxeter graph of F4 containing the Coxeter graph of F₄.[이미지참조] 97
Figure 6.11. The Coxeter graph of G2 containing the Coxeter graph of G₂.[이미지참조] 98
Figure 6.12. The Coxeter graph of I2(∞).[이미지참조] 98