Related papers: Systematic Enumeration of Fundamental Quantities I…
Compression of integer sets and sequences has been extensively studied for settings where elements follow a uniform probability distribution. In addition, methods exist that exploit clustering of elements in order to achieve higher…
We provide a combinatorial construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an arbitrary number of distinct…
We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where…
The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these…
We consider the question of computing the distribution of a permutation statistics over restricted permutations via enumeration schemes. The restricted permutations are those avoiding sets of vincular patterns (which include both classical…
In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in…
In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…
The Binary Jumbled String Matching problem is defined as: Given a string $s$ over $\{a,b\}$ of length $n$ and a query $(x,y)$, with $x,y$ non-negative integers, decide whether $s$ has a substring $t$ with exactly $x$ $a$'s and $y$ $b$'s.…
The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different…
In this paper, we study $F_{n}(x,k)$, the number of binary strings of length $n$ containing $x$ zeros and a longest subword of $k$ zeros. A recurrence relation for $F_{n}(x,k)$ is derived. We expressed few known numbers like Fibonacci,…
We investigate the longest common substring problem for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the…
We continue an earlier study, starting with unconstrained $n$-bitstrings, focusing now less on average behavior and more on uncertainty. The interplay between $\bullet$ longest runs of 0s and of 1s, when bitstrings are multus $\bullet$…
We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.
We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…
Getting tools that allow simple representations and comparisons of a set of categorical trajectories is of major interest for statisticians. Without loosing any information, we associate to each state a binary random indicator function,…
This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a functions of its summands as their number tends to infinity. The conditioning event is of moderate or…
In this paper we examine two ways of coding sequences in arithmetical theories. We investigate under what conditions they work. To be more precise, we study the creation of objects of a data-type that we call ur-strings, roughly sequences…
We present an algorithm for effectively generating binary sequences which would be rated by people as highly likely to have been generated by a random process, such as flipping a fair coin.
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a…
Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a…