A note on permutation regularity

The existence of a small partition of a combinatorial structure into random-like subparts, a so-called regular partition, has proven to be very useful in the study of extremal problems, and has deep algorithmic consequences. The main result in this direction is the Szemeredi Regularity Lemma in graph theory. In this note, we are concerned with regularity in permutations: we show that every permutation of a sufficiently large set has a regular partition into a small number of intervals. This refines the partition given by Cooper (2006) [10], which required an additional non-interval exceptional class. We also introduce a distance between permutations that plays an important role in the study of convergence of a permutation sequence. (C) 2011 Elsevier B.V. All rights reserved.

Design-based random permutation models with auxiliary information

We extend the random permutation model to obtain the best linear unbiased estimator of a finite population mean accounting for auxiliary variables under simple random sampling without replacement (SRS) or stratified SRS. The proposed method provides a systematic design-based justification for well-known results involving common estimators derived under minimal assumptions that do not require specification of a functional relationship between the response and the auxiliary variables.

ABSENCE OF STABILIZATION POINT OF THE SPECIES ACCUMULATION CURVE IN TROPICAL FORESTS

The definition of the sample size is a major problem in studies of phytosociology. The species accumulation curve is used to define the sampling sufficiency, but this method presents some limitations such as the absence of a stabilization point that can be objectively determined and the arbitrariness of the order of sampling units in the curve. A solution to this problem is the use of randomization procedures, e. g. permutation, for obtaining a mean species accumulation curve and empiric confidence intervals. However, the randomization process emphasizes the asymptotical character of the curve. Moreover, the inexistence of an inflection point in the curve makes it impossible to define objectively the point of optimum sample size.