Related papers: Enumeration of 3-letter patterns in compositions
Gessel's famous Bessel determinant formula gives the generating function of the number of permutations without increasing subsequences of a given length. Ekhad and Zeilberger proposed the challenge of finding a suitable generalization for…
In this paper we consider the palindromes that can be formed by taking unordered sets of $n$ elements from an alphabet of $b$ letters. In particular, we seek to find the probability that given a random member of this space we are able to…
We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We…
While recent generative models can produce engaging music, their utility is limited. The variation in the music is often left to chance, resulting in compositions that lack structure. Pieces extending beyond a minute can become incoherent…
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…
We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number…
Words in some natural languages can have a composite structure. Elements of this structure include the root (that could also be composite), prefixes and suffixes with which various nuances and relations to other words can be expressed.…
In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…
In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules.…
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…
An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in…
Recently, the general problem of enumerating permutations $\pi=\pi_1\cdots \pi_n$ such that $\pi_{i+r}-\pi_i \neq s$ for all $1\leq i\leq n-r$, where $r$ and $s$ are fixed, was considered by Spahn and Zeilberger. In this paper, we consider…
A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different…
We consider document listing on string collections, that is, finding in which strings a given pattern appears. In particular, we focus on repetitive collections: a collection of size $N$ over alphabet $[1,\sigma]$ is composed of $D$ copies…
In this article, dedicated with admiration and gratitude to guru Neil Sloane on his 75-th birthday, we observe that the generating functions for multi-set permutations that do not contain an increasing subsequence of length d, and where…
A Skolem sequence is a linear arrangement of the multiset, {1, 1, 2, 2, ..., n, n} such that if r in [n] appears in positions i and j, then |i-j| = r. We first translate the problem to a particular set of perfect matchings, then apply the…
In this paper, we study automatic keyphrase generation. Although conventional approaches to this task show promising results, they neglect correlation among keyphrases, resulting in duplication and coverage issues. To solve these problems,…
We consider the problem of enumerating the permutations containing exactly $k$ occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an…