Atnaujinkite slapukų nuostatas

Eulerian Numbers 1st ed. 2015 [Kietas viršelis]

5.00/5 (2 ratings by Goodreads)
  • Formatas: Hardback, 456 pages, aukštis x plotis: 235x155 mm, weight: 9198 g, 4 Illustrations, color; 74 Illustrations, black and white; XVIII, 456 p. 78 illus., 4 illus. in color., 1 Hardback
  • Serija: Birkhauser Advanced Texts / Basler Lehrbucher
  • Išleidimo metai: 13-Oct-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493930907
  • ISBN-13: 9781493930906
Kitos knygos pagal šią temą:
Eulerian Numbers 1st ed. 2015Didesnis paveikslėlis
  • Formatas: Hardback, 456 pages, aukštis x plotis: 235x155 mm, weight: 9198 g, 4 Illustrations, color; 74 Illustrations, black and white; XVIII, 456 p. 78 illus., 4 illus. in color., 1 Hardback
  • Serija: Birkhauser Advanced Texts / Basler Lehrbucher
  • Išleidimo metai: 13-Oct-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493930907
  • ISBN-13: 9781493930906
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781493930906.html
Raktažodžiai:
This text presents the Eulerian numbers in the context of modern enumerative, algebraic, and geometric combinatorics. The book first studies Eulerian numbers from a purely combinatorial point of view, then embarks on a tour of how these numbers arise in the study of hyperplane arrangements, polytopes, and simplicial complexes. Some topics include a thorough discussion of gamma-nonnegativity and real-rootedness for Eulerian polynomials, as well as the weak order and the shard intersection order of the symmetric group.

The book also includes a parallel story of Catalan combinatorics, wherein the Eulerian numbers are replaced with Narayana numbers. Again there is a progression from combinatorics to geometry, including discussion of the associahedron and the lattice of noncrossing partitions.

The final chapters discuss how both the Eulerian and Narayana numbers have analogues in any finite Coxeter group, with many of the same enumerative and geometric properties. Thereare four supplemental chapters throughout, which survey more advanced topics, including some open problems in combinatorial topology.

This textbook will serve a resource for experts in the field as well as for graduate students and others hoping to learn about these topics for the first time.

Recenzijos

This book is a beautiful modern collection about Eulerian numbers, which reflects its author's interest and expertise in the subject. The material is rich. Eulerian numbers and their close relatives, such as Narayana numbers, show up in a variety of topics from enumerative, algebraic, topological, and geometric combinatorics. It is self-contained and presented in a clear ... way. The author describes everything from the basics, so the text should be accessible to a broad audience interested in combinatorics. (Damir Yeliussizov, Mathematical Reviews, April, 2017)



This work offers wonderful material to supplement a course in combinatorics or discrete mathematics, or for a stimulating capstone course. Summing Up: Highly recommended. Lower-division undergraduates through researchers/faculty. (D. V. Feldman, Choice, Vol. 53 (11), July, 2016)

This book serves dual purposes. On the one hand, it is a monograph on Eulerian numbers and their generalizations. On the other hand, the book gives an introduction to contemporary enumerative, algebraic and geometric combinatorics, and it can be used as a text at beginning graduate or advanced undergraduate level. This is a well-written text for a good course. (Lįszló A. Székely, zbMATH 1337.05001, 2016)

Part I Combinatorics
1 Eulerian numbers
3(16)
1.1 Binomial coefficients
3(2)
1.2 Generating functions
5(1)
1.3 Classical Eulerian numbers
6(3)
1.4 Eulerian polynomials
9(1)
1.5 Two important identities
10(2)
1.6 Exponential generating function
12(2)
Problems
14(5)
2 Narayana numbers
19(28)
2.1 Catalan numbers
19(1)
2.2 Pattern-avoiding permutations
20(3)
2.3 Narayana numbers
23(3)
2.4 Dyck paths
26(8)
2.4.1 Counting all Dyck paths
27(2)
2.4.2 Counting Dyck paths by peaks
29(2)
2.4.3 A bijection with 231-avoiding permutations
31(3)
2.5 Planar binary trees
34(2)
2.6 Noncrossing partitions
36(4)
Problems
40(7)
3 Partially ordered sets
47(24)
3.1 Basic definitions and terminology
47(3)
3.2 Labeled posets and P-partitions
50(4)
3.3 The shard intersection order
54(3)
3.4 The lattice of noncrossing partitions
57(4)
3.5 Absolute order and Noncrossing partitions
61(3)
Problems
64(7)
4 Gamma-nonnegativity
71(24)
4.1 The idea of gamma-nonnegativity
71(1)
4.2 Gamma-nonnegativity for Eulerian numbers
72(4)
4.3 Gamma-nonnegativity for Narayana numbers
76(1)
4.4 Palindromicity, unimodality, and the gamma basis
77(3)
4.5 Computing the gamma vector
80(1)
4.6 Real roots and log-concavity
81(3)
4.7 Symmetric boolean decomposition
84(4)
Problems
88(7)
5 Weak order, hyperplane arrangements, and the Tamari lattice
95(32)
5.1 Inversions
95(3)
5.2 The weak order
98(2)
5.3 The braid arrangement
100(2)
5.4 Euclidean hyperplane arrangements
102(3)
5.5 Products of faces and the weak order on chambers
105(3)
5.6 Set compositions
108(5)
5.7 The Tamari lattice
113(2)
5.8 Rooted planar trees and faces of the associahedron
115(8)
Problems
123(4)
6 Refined enumeration
127(24)
6.1 The idea of a q-analogue
127(2)
6.2 Lattice paths by area
129(3)
6.3 Lattice paths by major index
132(2)
6.4 Euler-Mahonian distributions
134(3)
6.5 Descents and major index
137(2)
6.6 q-Catalan numbers
139(1)
6.7 q-Narayana numbers
140(3)
6.8 Dyck paths by area
143(6)
Problems
149(2)
7 Cubes, Carries, and an Amazing Matrix: (Supplemental)
151(12)
7.1 Slicing a cube
151(3)
7.2 Carries in addition
154(2)
7.3 The amazing matrix
156(7)
Part II Combinatorial topology
8 Simplicial complexes
163(22)
8.1 Abstract simplicial complexes
163(3)
8.2 Simple convex polytopes
166(1)
8.3 Boolean complexes
167(2)
8.4 The order complex of a poset
169(1)
8.5 Flag simplicial complexes
170(2)
8.6 Balanced simplicial complexes
172(1)
8.7 Face enumeration
173(2)
8.8 The h-vector
175(2)
8.9 The Dehn-Sommerville relations
177(4)
Problems
181(4)
9 Barycentric subdivision
185(18)
9.1 Barycentric subdivision of a finite cell complex
185(2)
9.2 The barycentric subdivision of a simplex
187(3)
9.3 Brenti and Welker's transformation
190(3)
9.4 The h-vector of sd(Δ) and j-Eulerian numbers
193(3)
9.5 Gamma-nonnegativity of h(sd(Δ))
196(4)
9.6 Real roots for barycentric subdivisions
200(1)
Problems
201(2)
10 Characterizing f-vectors: (Supplemental)
203(34)
10.1 Compressed simplicial complexes
203(4)
10.2 Proof of the compression lemma
207(8)
10.3 Kruskal-Katona-Schutzenberger inequalities
215(4)
10.4 Frankl-Furedi-Kalai inequalities
219(4)
10.5 Multicomplexes and M-vectors
223(2)
10.6 The Stanley-Reisner ring
225(3)
10.7 The upper bound theorem and the g-theorem
228(2)
10.8 Conjectures for flag spheres
230(7)
Part III Coxeter groups
11 Coxeter groups
237(36)
11.1 The symmetric group
237(6)
11.2 Finite Coxeter groups: generators and relations
243(3)
11.3 W-Mahonian distribution
246(1)
11.4 W-Eulerian numbers
246(4)
11.5 Finite reflection groups and root systems
250(7)
11.5.1 Type An--1
254(1)
11.5.2 Type Bn
254(1)
11.5.3 Type Cn
255(1)
11.5.4 Type Dn
255(1)
11.5.5 Roots for I2(m)
256(1)
11.6 The Coxeter arrangement and the Coxeter complex
257(2)
11.7 Action of W and cosets of parabolic subgroups
259(3)
11.8 Counting faces in the Coxeter complex
262(2)
11.9 The W-Euler-Mahonian distribution
264(2)
11.10 The weak order
266(3)
11.11 The shard intersection order
269(2)
Problems
271(2)
12 W-Narayana numbers
273(20)
12.1 Reflection length and Coxeter elements
273(3)
12.2 Absolute order and W-noncrossing partitions
276(1)
12.3 W-Catalan and W-Narayana numbers
277(3)
12.4 Coxeter-sortable elements
280(2)
12.5 Root posets and W-nonnesting partitions
282(5)
12.6 The W-associahedron
287(3)
Problems
290(3)
13 Combinatorics for Coxeter groups of types Bn and Dn: (Supplemental)
293(40)
13.1 Type Bn Eulerian numbers
293(4)
13.2 Type Bn gamma-nonnegativity
297(4)
13.3 Type Dn Eulerian numbers
301(2)
13.4 Type Dn gamma-nonnegativity
303(4)
13.5 Combinatorial models for shard intersections
307(13)
13.5.1 Type An--1
307(4)
13.5.2 Type Bn
311(5)
13.5.3 Type Dn
316(4)
13.6 Type Bn noncrossing partitions and Narayana numbers
320(5)
13.7 Gamma-nonnegativity for Cat(Bn; t)
325(2)
13.8 Type Dn noncrossing partitions and Narayana numbers
327(3)
13.9 Gamma-nonnegativity for Cat(Dn; t)
330(3)
14 Affine descents and the Steinberg torus: (Supplemental)
333(14)
14.1 Affine Weyl groups
333(1)
14.2 Faces of the affine Coxeter complex
334(4)
14.3 The Steinberg torus
338(3)
14.4 Affine Eulerian numbers
341(6)
14.4.1 Type An--1
341(1)
14.4.2 Type Bn
342(1)
14.4.3 Type Cn
343(1)
14.4.4 Type Dn
344(3)
References 347(12)
Hints and Solutions 359(94)
Index 453
T. Kyle Petersen is an Associate Professor of Mathematics at DePaul University, Chicago, USA.  His research areas include algebraic, enumerative, and topological combinatorics.  He received his PhD in Mathematics from Brandeis University.