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The coalescent with recombination is a fundamental model to describe the genealogical history of DNA sequence samples from recombining organisms. Considering recombination as a process which acts along genomes and which creates sequence segments with shared ancestry, we study the influence of single recombination events upon tree characteristics of the coalescent. We focus on properties such as tree height and tree balance and quantify analytically the changes in these quantities incurred by recombination in terms of probability distributions. We find that changes in tree topology are often relatively mild under conditions of neutral evolution, while changes in tree height are on average quite large. Our results add to a quantitative understanding of the spatial coalescent and provide the neutral reference to which the impact by other evolutionary scenarios, for instance tree distortion by selective sweeps, can be compared.

We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis.

A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.