54 research outputs found

    Unbalanced subtrees in binary rooted ordered and un-ordered trees

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    Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely unbalanced subtrees, where unbalancing is measured according to the so-called Colless's index. The size of the biggest unbalanced subtree becomes then a new parameter with respect to which we find several enumerations

    On the sub-permutations of pattern avoiding permutations

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    There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and probabilistic properties of sub-permutations and to investigate the relationships between 'local' and 'global' features using the concept of pattern avoidance. First, given a pattern {\mu}, we study how the avoidance of {\mu} in a permutation {\pi} affects the presence of other patterns in the sub-permutations of {\pi}. More precisely, considering patterns of length 3, we solve instances of the following problem: given a class of permutations K and a pattern {\mu}, we ask for the number of permutations ŌÄ‚ąąAvn(őľ)\pi \in Av_n(\mu) whose sub-permutations in K satisfy certain additional constraints on their size. Second, we study the probability for a generic pattern to be contained in a random permutation {\pi} of size n without being present in the sub-permutations of {\pi} generated by the entry 1‚ȧk‚ȧn1 \leq k \leq n. These theoretical results can be useful to define efficient randomized pattern-search procedures based on classical algorithms of pattern-recognition, while the general problem of pattern-search is NP-complete

    Coalescent histories for lodgepole species trees

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    Coalescent histories are combinatorial structures that describe for a given gene tree and species tree the possible lists of branches of the species tree on which the gene tree coalescences take place. Properties of the number of coalescent histories for gene trees and species trees affect a variety of probabilistic calculations in mathematical phylogenetics. Exact and asymptotic evaluations of the number of coalescent histories, however, are known only in a limited number of cases. Here we introduce a particular family of species trees, the \emph{lodgepole} species trees (őĽn)n‚Č•0(\lambda_n)_{n\geq 0}, in which tree őĽn\lambda_n has m=2n+1m=2n+1 taxa. We determine the number of coalescent histories for the lodgepole species trees, in the case that the gene tree matches the species tree, showing that this number grows with m!!m!! in the number of taxa mm. This computation demonstrates the existence of tree families in which the growth in the number of coalescent histories is faster than exponential. Further, it provides a substantial improvement on the lower bound for the ratio of the largest number of matching coalescent histories to the smallest number of matching coalescent histories for trees with mm taxa, increasing a previous bound of (ŌÄ/32)[(5m‚ąí12)/(4m‚ąí6)]mm(\sqrt{\pi} / 32)[(5m-12)/(4m-6)] m \sqrt{m} to [m‚ąí1/(4e)]m[ \sqrt{m-1}/(4 \sqrt{e}) ]^{m}. We discuss the implications of our enumerative results for phylogenetic computations

    On the number of ranked species trees producing anomalous ranked gene trees

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    Analysis of probability distributions conditional on species trees has demonstrated the existence of anomalous ranked gene trees (ARGTs), ranked gene trees that are more probable than the ranked gene tree that accords with the ranked species tree. Here, to improve the characterization of ARGTs, we study enumerative and probabilistic properties of two classes of ranked labeled species trees, focusing on the presence or avoidance of certain subtree patterns associated with the production of ARGTs. We provide exact enumerations and asymptotic estimates for cardinalities of these sets of trees, showing that as the number of species increases without bound, the fraction of all ranked labeled species trees that are ARGT-producing approaches 1. This result extends beyond earlier existence results to provide a probabilistic claim about the frequency of ARGTs

    On the maximal weight of (p,q)(p,q)-ary chain partitions with bounded parts

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    A (p,q)(p,q)-ary chain is a special type of chain partition of integers with parts of the form paqbp^aq^b for some fixed integers pp and qq. In this note, we are interested in the maximal weight of such partitions when their parts are distinct and cannot exceed a given bound mm. Characterizing the cases where the greedy choice fails, we prove that this maximal weight is, as a function of mm, asymptotically independent of max‚Ā°(p,q)\max(p,q), and we provide an efficient algorithm to compute it.Comment: 17 page

    A closed formula for the number of convex permutominoes

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    In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes
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