Related papers: Constructing Skolem sequences via generating trees
LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were used to encode ordinal trees and to…
Standard sequential generation methods assume a pre-specified generation order, such as text generation methods which generate words from left to right. In this work, we propose a framework for training models of text generation that…
We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and…
We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…
The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning…
String diagrams are a graphical language used to represent processes that can be composed sequentially or in parallel, which correspond graphically to horizontal or vertical juxtaposition. In this paper we demonstrate how to compute the…
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 closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also…
In this article, we give a precise mathematical meaning to `linear? time' that matches experimental behaviour of the algorithm. The sorting algorithm is not our own, it is a variant of radix sort with counting sort as a subroutine. The true…
We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences…
Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a…
We introduce several classes of pseudorandom sequences which represent a natural extension of classical methods in random number generation. The sequences are obtained from constructions on labeled binary trees, generalizing the well-known…
This project aims to investigate a novel sequence generation method inspired by the AlphaGo paradigm, adapting it for use with large language models (LLMs). The proposed approach involves creating search trees of different possible…
The transmission of a vertex in a connected graph is the sum of distances from that vertex to all the other vertices. A connected graph is transmission irregular if any two distinct vertices have different transmissions. We present an…
Most existing text generation models follow the sequence-to-sequence paradigm. Generative Grammar suggests that humans generate natural language texts by learning language grammar. We propose a syntax-guided generation schema, which…
A set A is a Sidon set in an additive group G if every element of G can be written at most one way as sum of two elements of A. A particular case of two-dimensional Sidon sets are the sonar sequences, which are two-dimensional…
Sequences have become first class citizens in supervised learning thanks to the resurgence of recurrent neural networks. Many complex tasks that require mapping from or to a sequence of observations can now be formulated with the…
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…
Text generation is a fundamental building block in natural language processing tasks. Existing sequential models performs autoregression directly over the text sequence and have difficulty generating long sentences of complex structures.…
The Stochastic Context Tree (SCOT) is a useful tool for studying infinite random sequences generated by an m-Markov Chain (m-MC). It captures the phenomenon that the probability distribution of the next state sometimes depends on less than…