54 research outputs found

### Unbalanced subtrees in binary rooted ordered and un-ordered trees

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

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 $\pi \in Av_n(\mu)$ 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 $1 \leq k \leq n$. 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

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 $(\lambda_n)_{n\geq 0}$, in which
tree $\lambda_n$ has $m=2n+1$ 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 $m!!$ in the
number of taxa $m$. 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 $m$ taxa,
increasing a previous bound of $(\sqrt{\pi} / 32)[(5m-12)/(4m-6)] m \sqrt{m}$
to $[ \sqrt{m-1}/(4 \sqrt{e}) ]^{m}$. We discuss the implications of our
enumerative results for phylogenetic computations

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

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 $(p,q)$-ary chain partitions with bounded parts

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

### A closed formula for the number of convex permutominoes

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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