3,356 research outputs found
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
Perfect Necklaces
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix
a finite alphabet. Recall that a word is a finite sequence of symbols in the
alphabet and a circular word, or necklace, is the equivalence class of a word
under rotations. For positive integers k and n, we call a necklace
(k,n)-perfect if each word of length k occurs exactly n times at positions
which are different modulo n for any convention on the starting point. We call
a necklace perfect if it is (k,k)-perfect for some k. We prove that every
arithmetic sequence with difference coprime with the alphabet size induces a
perfect necklace. In particular, the concatenation of all words of the same
length in lexicographic order yields a perfect necklace. For each k and n, we
give a closed formula for the number of (k,n)-perfect necklaces. Finally, we
prove that every infinite periodic sequence whose period coincides with some
(k,n)-perfect necklace for any n, passes all statistical tests of size up to k,
but not all larger tests. This last theorem motivated this work
Online version of the theorem of Thue
A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In
1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over
3 symbols. We consider the online variant of this result in which a
nonrepetitive sequence is constructed during a play between two players: Bob is
choosing a position in a sequence and Alice is inserting a symbol on that
position taken from a fixed set A. The goal of Bob is to force Alice to create
a repetition, and if he succeeds, then the game stops. The goal of Alice is
naturally to avoid that and thereby to construct a nonrepetitive sequence of
any given length. We prove that Alice has a strategy to play arbitrarily long
provided the size of the set A is at least 12. This is the online version of
the Theorem of Thue. The proof is based on nonrepetitive colorings of
outerplanar graphs. On the other hand, one can prove that even over 4 symbols
Alice has no chance to play for too long. The minimum size of the set of
symbols needed for the online version of Thue's theorem remains unknown
Analytic aspects of the shuffle product
There exist very lucid explanations of the combinatorial origins of rational
and algebraic functions, in particular with respect to regular and context free
languages. In the search to understand how to extend these natural
correspondences, we find that the shuffle product models many key aspects of
D-finite generating functions, a class which contains algebraic. We consider
several different takes on the shuffle product, shuffle closure, and shuffle
grammars, and give explicit generating function consequences. In the process,
we define a grammar class that models D-finite generating functions
The combinatorics of associated Hermite polynomials
We develop a combinatorial model of the associated Hermite polynomials and
their moments, and prove their orthogonality with a sign-reversing involution.
We find combinatorial interpretations of the moments as complete matchings,
connected complete matchings, oscillating tableaux, and rooted maps and show
weight-preserving bijections between these objects. Several identities,
linearization formulas, the moment generating function, and a second
combinatorial model are also derived.Comment: [v1]: 18 pages, 16 figures; presented at FPSAC 2007 [v2]: Some minor
errors fixed (thanks Bill Chen, Jang Soo Kim) and text rearranged and cleaned
up; no real content changes [v3]: fixed typos, to appear in European J.
Combinatoric
Intervals of Permutations with a Fixed Number of Descents are Shellable
The set of all permutations, ordered by pattern containment, is a poset. We
present an order isomorphism from the poset of permutations with a fixed number
of descents to a certain poset of words with subword order. We use this
bijection to show that intervals of permutations with a fixed number of
descents are shellable, and we present a formula for the M\"obius function of
these intervals. We present an alternative proof for a result on the M\"obius
function of intervals such that has exactly one descent. We
prove that if has exactly one descent and avoids 456123 and 356124, then
the intervals have no nontrivial disconnected subintervals; we
conjecture that these intervals are shellable
- …