50,693 research outputs found
Asymptotic behavior of some statistics in Ewens random permutations
The purpose of this article is to present a general method to find limiting
laws for some renormalized statistics on random permutations. The model
considered here is Ewens sampling model, which generalizes uniform random
permutations. We describe the asymptotic behavior of a large family of
statistics, including the number of occurrences of any given dashed pattern.
Our approach is based on the method of moments and relies on the following
intuition: two events involving the images of different integers are almost
independent.Comment: 32 pages: final version for EJP, produced by the author. An extended
abstract of 12 pages, published in the proceedings of AofA 2012, is also
available as version
Two Combinatorial Models with identical Statics yet different Dynamics
Motivated by the problem of sorting, we introduce two simple combinatorial
models with distinct Hamiltonians yet identical spectra (and hence partition
function) and show that the local dynamics of these models are very different.
After a deep quench, one model slowly relaxes to the sorted state whereas the
other model becomes blocked by the presence of stable local minima.Comment: 23 pages, 11 figure
Enumeration by kernel positions for strongly Bernoulli type truncation games on words
We find the winning strategy for a class of truncation games played on words.
As a consequence of the present author's recent results on some of these games
we obtain new formulas for Bernoulli numbers and polynomials of the second kind
and a new combinatorial model for the number of connected permutations of given
rank. For connected permutations, the decomposition used to find the winning
strategy is shown to be bijectively equivalent to King's decomposition, used to
recursively generate a transposition Gray code of the connected permutations
Self-organizing search lists using probabilistic back-pointers
A class of algorithms is given for maintaining self-organizing sequential search lists, where the only permutation applied is to move the accessed record of each search some distance towards the front of the list. During searches, these algorithms retain a back-pointer to a previously probed record in order to determine the destination of the accessed record's eventual move. The back-pointer does not traverse the list, but rather it is advanced occationally to point to the record just probed by the search algorithm. This avoids the cost of a second traversal through a significant portion of the list, which may be a significant savings when each record access may require a new page to be brought into primary memory. Probabilistic functions for deciding when to advance the pointer are presented and analyzed. These functions demonstrate average case complexities of measures such as asymptotic cost and convergence similar to some of the more common list update algorithms in the literature. In cases where the accessed record is moved forward a distance proportional to the distance to the front of the list, the use of these functions may save up to 50% of the time required for permuting the list
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