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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. There are 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.

Eulerian Numbers

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 B n and D n.- Affine Descents and the Steinberg Torus.- Hints and Solutions.