44,496 research outputs found
Generating permutations with a given major index
In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a
given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an
algorithm, which appears to be CAT, for generating permutations with a given
major index. In the present paper we give a new algorithm for generating a Gray
code for subexcedant sequences. We show that this algorithm is CAT and derive
it into a CAT generating algorithm for permutations with a given major index
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
On the growth rate of 1324-avoiding permutations
We give an improved algorithm for counting the number of -avoiding
permutations, resulting in 5 further terms of the generating function. We
analyse the known coefficients and find compelling evidence that unlike other
classical length-4 pattern-avoiding permutations, the generating function in
this case does not have an algebraic singularity. Rather, the number of
1324-avoiding permutations of length behaves as We estimate
and Comment: 20 pages, 10 figure
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur
SEALiP: A simple and efficient algorithm for listing permutation via starter set method
Algorithm for listing permutations for n elements is an arduous task.This paper attempts to introduce a novel method for generating permutations.The
fundamental concept for this method is to seek a starter set to begin with as an initial set to generate all distinct permutations. In order to demonstrate the algorithm, we are keen to list the permutations with the special references for cases of three and four objects.Based on this algorithm, a new method for listing permutations is developed and analyzed.This new permutation method will be compared with the existing lexicographic method.The results
revealed that this new method is more efficient in terms of computation time
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
Second-order preserving point process permutations
While random permutations of point processes are useful for generating counterfactuals in bivariate interaction tests, such permutations require that the underlying intensity be separable. In many real-world datasets where clustering or inhibition is present, such an assumption does not hold. Here, we introduce a simple combinatorial optimization algorithm that generates second-order preserving (SOP) point process permutations, for example, permutations of the times of events such that the function of the permuted process matches the function of the data. We apply the algorithm to synthetic data generated by a self-exciting Hawkes process and a self-avoiding point process, along with data from Los Angeles on earthquakes and arsons and data from Indianapolis on law enforcement drug seizures and overdoses. In all cases, we are able to generate a diverse sample of permuted point processes where the distribution of the functions closely matches that of the data. We then show how SOP point process permutations can be used in two applications: (1) bivariate Knox tests and (2) data augmentation to improve deep learning-based space-time forecasts
- …