We study the problem of constructing infinite words having a prescribed finite set P of palindromes. We first establish that the language of all words with palindromic factors in P is rational. As a consequence we derive that there exists, with some additional mild condition, infinite words having P as palindromic factors. We prove that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition. Asymmetric words, those that are not the product of two palindromes, play a fundamental role and an enumeration is provided.

We describe algorithms to study the space of all possible reconciliations between a gene tree and a species tree, that is counting the size of this space, uniformly generate a random reconciliation, and exploring this space in optimal time using combinatorial operators. We also extend these algorithms for optimal and sub-optimal reconciliations according to the three usual combinatorial costs (duplication, loss, and mutation). Applying these algorithms to simulated and real gene family evolutionary scenarios, we observe that the LCA (Last Common Ancestor) based reconciliation is almost always identical to the real one.

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The results of our simulations suggest that, with exact gene trees obtained by a simple birth-and-death process and realistic gene duplication/loss rates, a very small subset of all reconciliations needs to be explored in order to approximate very closely the posterior probability of the most likely reconciliations. For cases where the posterior probability mass is more evenly dispersed, our method allows to explore efficiently the required subspace of reconciliations.