Eulerian Numbers
Seiten
2015
|
1st ed. 2015
Springer-Verlag New York Inc.
978-1-4939-3090-6 (ISBN)
Springer-Verlag New York Inc.
978-1-4939-3090-6 (ISBN)
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.
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.
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.
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.
Eulerian Numbers.- Narayana Numbers.- Partially Ordered Sets.- Gamma-nonnegativity.- Weak Order, Hyperplane Arrangements, and the Tamari Lattice.- Refined Enumeration.- Simplicial Complexes.- Barycentric Subdivision.- Coxeter Groups.- W-Narayana Numbers.- Cubes, Carries, and an Amazing Matrix.- Characterizing f-vectors.- Combinatorics for Coxeter groups of Types Bn and Dn.- Affine Descents and the Steinberg Torus.- Hints and Solutions.
| Reihe/Serie | Birkhäuser Advanced Texts Basler Lehrbücher |
|---|---|
| Zusatzinfo | 4 Illustrations, color; 74 Illustrations, black and white; XVIII, 456 p. 78 illus., 4 illus. in color. |
| Verlagsort | New York |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Mathematik / Informatik ► Mathematik ► Graphentheorie | |
| Schlagworte | Catalan numbers • Coxeter groups • Eulerian numbers • Gal's conjecture • gamma-nonnegativity • Gessel's conjecture • Narayana numbers • simplicial complex |
| ISBN-10 | 1-4939-3090-7 / 1493930907 |
| ISBN-13 | 978-1-4939-3090-6 / 9781493930906 |
| Zustand | Neuware |
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