Related papers: Generating sequences and key polynomials
Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infinite family of scattered sequences of order three is obtained. Equivalence…
This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
We present a conditional text generation framework that posits sentential expressions of possible causes and effects. This framework depends on two novel resources we develop in the course of this work: a very large-scale collection of…
By linking conceptual theories with observed data, generative models can support reasoning in complex situations. They have come to play a central role both within and beyond statistics, providing the basis for power analysis in molecular…
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…
We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we…
A new method for parallel generation of $q$-valued pseudorandom sequence based on the presentation of systems generating logical formulae by means of arithmetic polynomials is proposed. Fragment consisting of $k$-elements of $q$-valued…
We study a class polynomials obtained from an enumeration of the number of queen paths. In particular, we find the generating function for the diagonal sequence of this table and the zero distribution of a sequence of related polynomials.
We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently…
Defining words in a textual context is a useful task both for practical purposes and for gaining insight into distributed word representations. Building on the distributional hypothesis, we argue here that the most natural formalization of…
The purpose of this paper is twofold; (1) to develop several identities for the Generalized $k$-Pell sequence (including those of Binet, Catalan, Cassini, and d'Ocagne), and (2) to study applications of tridiagonal generating matrices for…
Sequence generation applications require satisfying semantic constraints, such as ensuring that programs are correct, using certain keywords, or avoiding undesirable content. Language models, whether fine-tuned or prompted with few-shot…
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 paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.
This paper introduces a unified theoretical perspective that views deep generative models as probability transformation functions. Despite the apparent differences in architecture and training methodologies among various types of generative…
Automatic generation of sequences has been a highly explored field in the last years. In particular, natural language processing and automatic music composition have gained importance due to the recent advances in machine learning and…
Recent advances in neural network-based generative modeling have reignited the hopes in having computer systems capable of seamlessly conversing with humans and able to understand natural language. Neural architectures have been employed to…
Diese kurze Einfuehrung in Theorie und Berechnung linearer Rekurrenzen versucht, eine Luecke in der Literatur zu fuellen. Zu diesem Zweck sind viele ausfuehrliche Beispiele angegeben. This short introduction to theory and usage of linear…
The main object of the paper is to reveal connections between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials introduced by Catalan. This is achieved by relating the respective (ordinary and exponential)…