211,651 research outputs found
Forming Sequences of Patterns with Luminous Robots
The extensive studies on computing by a team of identical mobile robots operating in the plane in Look-Compute-Move cycles have been carried out mainly in the traditional {mathcal{ OBLOT}} model, where the robots are silent (have no communication capabilities) and oblivious (in a cycle, they have no memory previous cycles). To partially overcome the limits of obliviousness and silence while maintaining some of their advantages, the stronger model of luminous robots, {mathcal{ LUMI}} , has been introduced where the robots, otherwise oblivious and silent, carry a visible light that can take a number of different colors; a color can be seen by observing robots, and persists from a cycle to the next. In the study of the computational impact of lights, an immediate concern has been to understand and determine the additional computational strength of {mathcal{ LUMI}} over {mathcal{ OBLOT}}. Within this line of investigation, we examine the problem of forming a sequence of geometric patterns, PatternSequenceFormation. A complete characterization of the sequences of patterns formable from a given starting configuration has been determined in the {mathcal{ OBLOT}} model. In this paper, we study the formation of sequences of patterns in the {mathcal{ LUMI}} model and provide a complete characterization. The characterization is constructive: our universal protocol forms all formable sequences, and it does so asynchronously and without rigidity. This characterization explicitly and clearly identifies the computational strength of {mathcal{ LUMI}} over {mathcal{ OBLOT}} with respect to the Pattern Sequence Formation problem
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
On small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
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