1,268 research outputs found
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
Universal cycles for k-subsets of an n-set
Generalized from the classic de Bruijn sequence, a universal cycle is a compact cyclic list of information. Existence of universal cycles has been established for a variety of families of combinatorial structures. These results, by encoding each object within a combinatorial family as a length-j word, employ a modified version of the de Bruijn graph to establish a correspondence between an Eulerian circuit and a universal cycle.
We explore the existence of universal cycles for k-subsets of the integers {1, 2,...,n}. The fact that sets are unordered seems to prevent the use of the established encoding techniques used in proving existence. We explore this difficulty and introduce an intermediate step that may allow us to use the familiar encoding and correspondence to prove existence.
Moreover, mathematicians Persi Diaconis and Ron Graham hold that the construction of universal cycles has proceeded by clever, hard, ad-hoc arguments and that no general theory exists. Accordingly, our work pushes for a more general approach that can inform other universal cycle problems
On Universal Cycles for new Classes of Combinatorial Structures
A universal cycle (u-cycle) is a compact listing of a collection of
combinatorial objects. In this paper, we use natural encodings of these objects
to show the existence of u-cycles for collections of subsets, matroids,
restricted multisets, chains of subsets, multichains, and lattice paths. For
subsets, we show that a u-cycle exists for the -subsets of an -set if we
let vary in a non zero length interval. We use this result to construct a
"covering" of length for all subsets of of size
exactly with a specific formula for the term. We also show that
u-cycles exist for all -length words over some alphabet which
contain all characters from Using this result we provide
u-cycles for encodings of Sperner families of size 2 and proper chains of
subsets
Universal Lyndon Words
A word over an alphabet is a Lyndon word if there exists an
order defined on for which is lexicographically smaller than all
of its conjugates (other than itself). We introduce and study \emph{universal
Lyndon words}, which are words over an -letter alphabet that have length
and such that all the conjugates are Lyndon words. We show that universal
Lyndon words exist for every and exhibit combinatorial and structural
properties of these words. We then define particular prefix codes, which we
call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in
bijection with the set of the shortest unrepeated prefixes of the conjugates of
a universal Lyndon word. This allows us to give an algorithm for constructing
all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201
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