Related papers: Counting occurrences of some subword patterns
Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain…
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are…
We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
In this paper, we derive exact formulas for generating functions counting the number of $n$-ary words avoiding strictly increasing subwords of length $k$, and provide some applications of these formulas.
Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…
An avoidance pattern where the letters within an occurrence of which are required to be adjacent is referred to as a subword. In this paper, we enumerate members of the set NC_n of non-crossing partitions of length n according to the number…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
This is a compendium of generating functions involving single, double sums and definite integrals. These generating functions also involve special functions in both the summand function and closed form solution.
In this note we describe a new method of counting the number of unordered factorizations of a natural number by means of a generating function and a recurrence relation arising from it, which improves an earlier result in this direction.
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a…
We find exact formulas and/or generating functions for the number of words avoiding 3-letter generalized multipermutation patterns and find which of them are equally avoided.
Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all…
In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the…