Related papers: Constructing Skolem sequences via generating trees
This paper describes a design that can be used for Explainable AI. The lower level is a nested ensemble of patterns created by self-organisation. The upper level is a hierarchical tree, where nodes are linked through individual concepts, so…
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…
The number of spanning trees in a class of directed circulant graphs with generators depending linearly on the number of vertices $\beta n$, and in the $n$-th and $(n-1)$-th power graphs of the $\beta n$-cycle are evaluated as a product of…
We apply the method of spectral sequences to study classical problems in analysis. We illustrate the method by finding polynomial vector fields that commute with a given polynomial vector field and finding integrals of polynomial…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
The score of a vertex $x$ in an oriented graph is defined to be its outdegree, \emph{i.e.}, the number of arcs with initial vertex $x$. The score sequence of an oriented graph is the sequence of all scores arranged in nondecreasing order.…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…
When all the elements of the multiset $\{1,1,2,2,3,3,\ldots,k,k\}$ are placed in the cells of a $2\times k$ rectangular array, in how many configurations are exactly $v$ of the pairs directly over top one another, and exactly $h$ directly…
For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every…
Sequence-to-sequence models have been applied to the conversation response generation problem where the source sequence is the conversation history and the target sequence is the response. Unlike translation, conversation responding is…
We describe those complete linearly ordered topological spaces $X$ which are homogeneous (=CHLOTS). That is, $X$ is order isomorphic with any nonempty open interval in $X$. Using countable tail-like ordinals as indices, we build towers of…
Classic cycle-joining techniques have found widespread application in creating universal cycles for a diverse range of combinatorial objects, such as shorthand permutations, weak orders, orientable sequences, and various subsets of $k$-ary…
Traces and their extension called combined traces (comtraces) are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are…
Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer larger than $a_m$ that can be written as the sum of two (not necessarily distinct)…
We define a notion of higher order renormalization group equation and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a $k$th order renormalization group equation when…
Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by…
The focus of this work is to show how to combine Zech's logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences of any order. A basic implementation is supplied as a proof-of-concept.…
We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the…
This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any n-tuple occurs at most once in a period in either direction. As has been previously described, such…
Random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging. Random forests are known for their good practical performance, particularly in high dimensional set-tings. On the theoretical…