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.

Higher identities for the ternary commutator

We use computer algebra to study polynomial identities for the trilinear operation [a, b, c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a, b, c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11. The only irreducible representations of dimension <400 with new identities correspond to partitions 2(5), 1 and 2(4), 1(3) and have dimensions 132 and 165. We construct an explicit new multilinear identity for partition 2(5), 1 and we demonstrate the existence of a new non-multilinear identity in which the underlying variables are permutations of a(2)b(2)c(2)d(2)e(2) f.

Diversity and genetic structure in natural populations of Hancornia speciosa var. speciosa Gomes in northeastern Brazil

Hancornia speciosa Gomes is a fruit tree native from Brazil that belongs to Apocinaceae family, and is popularly known as Mangabeira. Its fruits are widely consumed raw or processed as fruit jam, juices and ice creams, which have made it a target of intense exploitation. The extractive activities and intense human activity on the environment of natural occurrence of H. speciosa has caused genetic erosion in the species and little is known about the ecology or genetic structure of natural populations. The objective of this research was the evaluation of the genetic diversity and genetic structure of H. speciosa var. speciosa. The genetic variability was assessed using 11 allozyme loci with a sample of 164 individuals distributed in six natural populations located in the States of Pernambuco and Alagoas, Northeastern Brazil. The results showed a high level of genetic diversity within the species (e= 0.36) seeing that the most of the genetic variability of H. speciosa var. speciosa is within its natural populations with low difference among populations ( or = 0.081). The inbreeding values within ( = -0.555) and among populations ( =-0.428) were low showing lacking of endogamy and a surplus of heterozygotes. The estimated gene flow ( m ) was high, ranging from 2.20 to 13.18, indicating to be enough to prevent the effects of genetic drift and genetic differentiation among populations. The multivariate analyses indicated that there is a relationship between genetic and geographical distances, which was confirmed by a spatial pattern analysis using Mantel test (r = 0.3598; p = 0.0920) with 1000 random permutations. The high genetic diversity index in these populations indicates potential for in situ genetic conservation.