Related papers: Generating function for Naturalized Series: The ca…
We reprove the countable splitting lemma by adapting Nawrotzki's algorithm which produces a sequence that converges to a solution. Our algorithm combines Nawrotzki's approach with taking finite cuts. It is constructive in the sense that…
Recently, a plethora of works have proposed inference-time algorithms (e.g. best-of-n), which incorporate verifiers to assist the generation process. Their quality-efficiency trade-offs have been empirically benchmarked on a variety of…
The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a…
We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and in…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…
As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces:…
The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that…
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(\l_1, \l_2, \dots, \l_k) \in \Z_{> 0}^k$ such that $n = \l_1 + \l_2 + \dots + \l_k$. Our goal is to enumerate those compositions whose parts $\l_1, \l_2, \dots, \l_k$ avoid a…
Many language generation tasks require the production of text conditioned on both structured and unstructured inputs. We present a novel neural network architecture which generates an output sequence conditioned on an arbitrary number of…
In recent years, the generation of conversation content based on deep neural networks has attracted many researchers. However, traditional neural language models tend to generate general replies, lacking logical and emotional factors. This…
In this paper we propose an algorithm to generate binary words with no more 0's than 1's having a fixed number of 1's and avoiding the pattern $(10)^j1$ for any fixed $j \geq 1$. We will prove that this generation is exhaustive, that is,…
One of the challenges in text generation is to control text generation as intended by the user. Previous studies proposed specifying the keywords that should be included in the generated text. However, this approach is insufficient to…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also…
Building upon the rule-algebraic stochastic mechanics framework, we present new results on the relationship of stochastic rewriting systems described in terms of continuous-time Markov chains, their embedded discrete-time Markov chains and…
In this paper, we prove results on enumerations of sets of Rota-Baxter words in a finite number of generators and a finite number of unary operators. Rota-Baxter words are words formed by concatenating generators and images of words under…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of "elementary operations" on positive integral sequences is introduced. Our main theorem answers the question of…
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…
Generative commonsense reasoning is the capability of a language model to generate a sentence with a given concept-set that is based on commonsense knowledge. However, generative language models still struggle to provide outputs, and the…