arXiv subjects
Publications and source records attributed to Antonio Bernini.
We study k-coloured Motzkin paths, namely Motzkin paths in which horizontal steps can be coloured in k different ways, and investigate their connection with the number of prefixes ending at odd height from both an analytical and a combinatorial point of view. Moreover, the combinatorial approach provides a random generation algorithm for k-coloured Motzkin paths in linear-time.
We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this new mex statistic for several classical combinatorial structures, by providing the mex generating function and/or a closed formula for its coefficients in each of the cases.
Catalan numbers and their interpretations in terms of Dyck paths are widely used in different topics of applied mathematics and computer science. Here, we consider a general approach for constrained Dyck paths. In particular, we study Dyck paths of height at most $h$ with the additional restriction of having no $k-1$ consecutive valleys at height $h-1$. We give a combinatorial description of this class of paths and derive enumeration formulas using classical techniques for counting constrained lattice paths. As a consequence of this analysis, we obtain an identity involving Catalan numbers which, to the best of the authors' knowledge, does not appear in the existing literature. This identity arises naturally from the combinatorial interpretation and provides a new relation among families of Dyck paths with height and local structural constraints.
Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational Dyck paths, and provide the associated generating function, according to their semilength, as well as the construction of such a class. Moreover, we characterize some subsets of the rational Dyck paths that are enumerated by the $\mathbb Q$-bonacci numbers.
We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called Q-bonacci numbers (recently introduced by Kirgizov) which generalize the classical q-bonacci numbers in the case where q is a positive rational.
We propose a method for the construction of sets of variable dimension strong non-overlapping matrices basing on any strong non-overlapping set of strings.
Dyck paths having height at most $h$ and without valleys at height $h-1$ are combinatorially interpreted by means of 312-avoding permutations with some restrictions on their \emph{left-to-right maxima}. The results are obtained by analyzing a restriction of a well-known bijection between the sets of Dyck paths and 312-avoding permutations. We also provide a recursive formula enumerating these two structures using ECO method and the theory of production matrices. As a further result we obtain a family of combinatorial identities involving Catalan numbers.
Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences leading to the binary strings avoiding $1^k$. We prove a bijection between the set %$F_n^{(k)}$ of strings of length $n$ and the set of permutations of $S_{n+1}(321,312,23\ldots(k+1)1)$. Finally, basing on a known Gray code for those strings, we define a Gray code for $S_{n+1}(321,312,23\ldots(k+1)1)$, where two consecutive permutations differ by an adjacent transposition.
Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive and random generation.
Pulmonary embolism (PE) and deep vein thrombosis (DVT) are gathered in venous thromboembolism (VTE) and represent the third cause of cardiovascular diseases. Recent studies suggest that meteorological parameters as atmospheric pressure, temperature, and humidity could affect PE incidence but, nowadays, the relationship between these two phenomena is debated and the evidence is not completely explained. The clinical experience of the Department of Emergency Medicine at AOUC Hospital suggests the possibility that a relationship effectively exists. We have collected data concerning the Emergency Medicine Unit admissions of PE patients to confirm our hypothesis. At the same time, atmospheric parameters are collected from the Lamma Consortium of Tuscany region. We have implemented new IT models and statistic tools by using semi-hourly records of weather time high resolution data to process the dataset. We have carried out tools from econometrics, like mobile means, and we have studied anomalies through the search for peaks and possible patterns. We have created a framework in Python to represent and study time series and to analyze data and plot graphs. The project has been uploaded on GitHub. Our analyses highlighted a strong correlation between the moving averages of atmospheric pressure and those of the hospitalizations number (R= -0.9468, p<0,001) although causality is still unknown. The existence of an increase in the number of hospitalizations in the days following short-to-medium periods of time characterized by a high number of half-hourly pressure changes is also detected. The spectrograms studies obtained by the Fourier transform requires to increase the dataset. The analyzed data (especially hospitalization data) were too few to carry out this kind of analyses.
Since some years, non-overlapping sets of strings (also called cross-bifix-free sets) have had an increasing interest in the frame of the researches about Theory of Codes. Recently some non-overlapping sets of strings with variable length were introduced. Moreover, the notion of non-overlapping strings has been naturally extended to the two dimensional case leading to several definitions of non-overlapping sets of matrices (or pictures). Starting from these results, in this paper we introduce non-overlapping sets of binary matrices having variable dimension. Indeed, we use non-overlapping variable length strings as rows of the matrices and imposing the avoidance of two consecutive patterns of length k, we get the desired sets of non-ovelapping matrices with variable dimension.
We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.
We define a set of binary matrices where any two of them can not be placed one on the other in a way such that the corresponding entries coincide. The rows of the matrices are obtained by means of Dyck words. The cardinality of the set of such matrices involves Catalan numbers.
Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the $k$-generalized Fibonacci numbers. Moreover, the generating function for the enumerating sequence is easily seen to be rational.
We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $σ\leq τ$ whenever the permutation $τ$ contains an occurrence of the permutation $σ$ in which all the entries are adjacent in $τ$ except at most the first and the second. We investigate the Möbius function of the quasi-consecutive pattern poset and we completely determine it for those intervals $[σ,τ]$ such that $σ$ occurs precisely once in $τ$.
A bidimensional bifix (in short bibifix) of a square matrix T is a square submatrix of T which occurs in the top-left and bottom-right corners of T. This allows us to extend the definition of bifix-free words and cross-bifix-free set of words to bidimensional structures. In this paper we exhaustively generate all the bibifix-free square matrices and we construct a particular non-expandable cross-bibifix-free set of square matrices. Moreover, we provide a Gray code for listing this set.
A cross-bifix-free set of words is a set in which no prefix of any length of any word is the suffix of any other word in the set. A construction of cross-bifix-free sets has recently been proposed by Chee {\it et al.} in 2013 within a constant factor of optimality. We propose a \emph{trace partitioned} Gray code for these cross-bifix-free sets and a CAT algorithm generating it.
We provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.