On products of long cycles: short cycle dependence and separation probabilities

We present various results on multiplying cycles in the symmetric group. One result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on n symbols as a product of an n-cycle and an (n - 1)-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths. Another application is related to the recent notion of separation probabilities for permutations introduced by Bernardi, Du, Morales and Stanley (2014)

Genetic Clustering of Hepatitis C Virus Strains and Severity of Recurrent Hepatitis after Liver Transplantation

The influence of viral factors on the severity of hepatitis C virus (HCV)-related liver disease is controversial. We studied 68 liver transplant patients with recurrent hepatitis C, of whom 53 were infected by genotype 1 strains. Relationships between core sequences, serum HCV RNA levels, and fibrosis scores for each patient were analyzed in pairwise fashion 5 years after transplantation. We used Mantel's test, a matrix correlation method, to evaluate the correspondence between measured genetic distances and observed phenotypic differences. No clear relationship was found when all 68 patients were analyzed. In contrast, when the 53 patients infected by genotype 1 strains were analyzed, a strong positive relationship was found between genetic distance and differences in 5-year fibrosis scores (P = 0.001) and differences in virus load (P = 0.009). In other words, the smaller the genetic distance between two patients' viral core sequences, the smaller the difference between the two patients' fibrosis scores and viral replication levels. No relationship was found between genetic distance and differences in age, sex, or immunosuppression. In multivariate analysis, the degree of fibrosis was negatively related to the virus load (r = −0.68; P = 0.003). In the particular setting of liver transplantation, and among strains with closely related phylogenetic backgrounds (genotype 1), this study points to a correlation between the HCV genetic sequence and the variability of disease expression