Search results for "Gray code"
showing 10 items of 32 documents
Combinatorial Gray codes for classes of pattern avoiding permutations
2007
The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schr\"oder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\s_n(\tau)$ for all $\tau\in \s_3$ and the obtained Gray codes have distances 4 and 5.
More restrictive Gray codes for necklaces and Lyndon words
2008
In the last years, the order induced by the Binary Reflected Gray Code or its generalizations shown an increasing interest. In this note we show that the BRGC order induces a cyclic 2-Gray code on the set of binary necklaces and Lyndon words and a cyclic 3-Gray code on the unordered counterparts. This is an improvement and a generalization to unlabeled words of the result in [V. Vajnovszki, Gray code order for Lyndon words, Discrete Math. Theoret. Comput. Sci. 9 (2) (2007) 145-152; M. Weston, V. Vajnovszki, Gray codes for necklaces and Lyndon words of arbitrary base, Pure Mathematics and Applications/Algebra and Theoretical Computer Science, in press]; however an algorithmic implementation …
Gray coding cubic planar maps
2016
International audience; The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps. (C) 2015 Elsevier B.V. All rights reserved.
Some Generalizations of a Simion Schmidt Bijection
2007
In 1985, Simion and Schmidt gave a constructive bijection φ from the set of all length (n-1) binary strings having no two consecutive 1s to the set of all length n permutations avoiding all patterns in {123,132,213}. In this paper, we generalize φ to an injective function from {0,1}n-1 to the set Sn of all length n permutations and derive from it four bijections φ : P →Q where P⊆{0,1}n-1 and Q ⊂ Sn. The domains are sets of restricted binary strings and the codomains are sets of pattern-avoiding permutations. As a particular case we retrieve the original Simion–Schmidt bijection. We also show that the bijections obtained are actually combinatorial isomorphisms, i.e. closeness-preserving bije…
Super Resolution Methods Implementing Diffractive Masks Having a Certain Degree of Periodicity
2011
This section presents an approach that provides super resolved imaging at the center of the field of view and yet allows to see the remaining of the original field of view with original resolution. This operation resembles optical zooming while the zoomed and the nonzoomed images are obtained simultaneously. This is obtained by taking a single snap-shot and using a single imaging lens. The technique utilizes a special static/still coding element and a postprocessing algorithmic, without any mechanical movements.
A loop-free two-close Gray-code algorithm for listing k-ary Dyck words
2006
AbstractP. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close—that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at least k−1 times as many 0's as 1's, we obtain k-suffixes: suffixes of k-ary Dyck words. Combinations are retrieved as special case by setting k=1 and k-ary Dyck words are retrieved as a special case by imposing the additional condition that the entire string has exactly k−1 times as many 0's a…
The pure descent statistic on permutations
2017
International audience; We introduce a new statistic based on permutation descents which has a distribution given by the Stirling numbers of the first kind, i.e., with the same distribution as for the number of cycles in permutations. We study this statistic on the sets of permutations avoiding one pattern of length three by giving bivariate generating functions. As a consequence, new classes of permutations enumerated by the Motzkin numbers are obtained. Finally, we deduce results about the popularity of the pure descents in all these restricted sets. (C) 2017 Elsevier B.V. All rights reserved.
Whole mirror duplication-random loss model and pattern avoiding permutations
2010
International audience; In this paper we study the problem of the whole mirror duplication-random loss model in terms of pattern avoiding permutations. We prove that the class of permutations obtained with this model after a given number p of duplications of the identity is the class of permutations avoiding the alternating permutations of length p2+1. We also compute the number of duplications necessary and sufficient to obtain any permutation of length n. We provide two efficient algorithms to reconstitute a possible scenario of whole mirror duplications from identity to any permutation of length n. One of them uses the well-known binary reflected Gray code (Gray, 1953). Other relative mo…
Gray code for Cayley permutations
2003
A length-n Cayley permutation p of a total ordered set S is a length-n sequence of elements from S, subject to the condition that if an element x appears in p then all elements y < x also appear in p . In this paper, we give a Gray code list for the set of length-n Cayley permutations. Two successive permutations in this list differ at most in two positions.
Gray code for compositions of n with parts 1 and p
2009
International audience