27,765 research outputs found
The algebraic combinatorics of snakes
Snakes are analogues of alternating permutations defined for any Coxeter
group. We study these objects from the point of view of combinatorial Hopf
algebras, such as noncommutative symmetric functions and their generalizations.
The main purpose is to show that several properties of the generating functions
of snakes, such as differential equations or closed form as trigonometric
functions, can be lifted at the level of noncommutative symmetric functions or
free quasi-symmetric functions. The results take the form of algebraic
identities for type B noncommutative symmetric functions, noncommutative
supersymmetric functions and colored free quasi-symmetric functions.Comment: 29 pages, Late
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Asymmetric function theory
The classical theory of symmetric functions has a central position in
algebraic combinatorics, bridging aspects of representation theory,
combinatorics, and enumerative geometry. More recently, this theory has been
fruitfully extended to the larger ring of quasisymmetric functions, with
corresponding applications. Here, we survey recent work extending this theory
further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the
Schubert calculus conference in Guangzhou, Nov. 201
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