1,081 research outputs found
On Universal Cycles for Multisets
A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of
integers from [n] with the property that each t-multiset of
[n] appears exactly once consecutively in the sequence. For such a sequence to
exist it is necessary that n divides , and it is reasonable
to conjecture that this condition is sufficient for large enough n in terms of
t. We prove the conjecture completely for t in {2,3} and partially for t in
{4,6}. These results also support a positive answer to a question of Knuth.Comment: 14 pages, two figures, will appear in Discrete Mathematics' special
issue on de Bruijn Cycles, Gray Codes and their generalizations; paper
revised according to journal referees' suggestion
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
Limits on P Systems with Proteins and Without Division
In the field of Membrane Computing, computational complexity theory has
been widely studied trying to nd frontiers of efficiency by means of syntactic or semantical ingredients. The objective of this is to nd two kinds of systems, one non-efficient
and another one, at least, presumably efficient, that is, that can solve NP-complete prob-
lems in polynomial time, and adapt a solution of such a problem in the former. If it is
possible, then P = NP. Several borderlines have been defi ned, and new characterizations
of different types of membrane systems have been published.
In this work, a certain type of P system, where proteins act as a supporting element
for a rule to be red, is studied. In particular, while division rules, the abstraction of
cellular mitosis is forbidden, only problems from class P can be solved, in contrast to the
result obtained allowing them.Ministerio de EconomÃa y Competitividad TIN2017-89842-PNational Natural Science Foundation of China No 6132010600
On the Parikh-de-Bruijn grid
We introduce the Parikh-de-Bruijn grid, a graph whose vertices are
fixed-order Parikh vectors, and whose edges are given by a simple shift
operation. This graph gives structural insight into the nature of sets of
Parikh vectors as well as that of the Parikh set of a given string. We show its
utility by proving some results on Parikh-de-Bruijn strings, the abelian analog
of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl
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