Related papers: Counting occurrences of some subword patterns
We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…
We use generating functions to enumerate Arndt compositions, that is, integer compositions where there is a descent between every second pair of parts, starting with the first and second part, and so on. In 2013, J\"org Arndt noted that…
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius…
In this paper we find the generating function for the number of vertices that have k elements in their subtree and use this generating function to calculate the probability that a vertex has a size k subtree. We also show how this same…
The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed…
There are two methods for counting the number of occurrences of a string in another large string. One is to count the number of places where the string is found. The other is to determine how many pieces of string can be extracted without…
Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern $123$ exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern $123$ exactly…
Keyphrase generation refers to the task of producing a set of words or phrases that summarises the content of a document. Continuous efforts have been dedicated to this task over the past few years, spreading across multiple lines of…
We address the problem of counting the number of strings in a collection where a given pattern appears, which has applications in information retrieval and data mining. Existing solutions are in a theoretical stage. We implement these…
In this paper we provide an overview of a series of recent results regarding algorithms for searching for subsequences in words or for the analysis of the sets of subsequences occurring in a word.
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…
We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…
We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter \alpha = -1. We describe how such a series can be…
We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…
The distributions of the number of occurrences of words (the distributions of words for short) play key roles in information theory, statistics, probability theory, ergodic theory, computer science, and DNA analysis. Bassino et al. 2010 and…
We introduce data structures answering queries concerning the occurrences of patterns from a given dictionary $\mathcal{D}$ in fragments of a given string $T$ of length $n$. The dictionary is internal in the sense that each pattern in…
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde…
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a…