Related papers: On Number Representation
In this paper, we give a new representation of the Fibonacci numbers. This is achieved using Fibonacci trees. With the help of this representation, the nth Fibonacci number can be calculated without having any knowledge about the previous…
Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a…
A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random…
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…
Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance…
It is a long-standing question of Stanley whether or not the chromatic symmetric function (CSF) distinguishes unrooted trees. Previously, the best computational result, due to Russell, proved that it distinguishes all trees with at most…
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
Every encoding has priori information if the encoding represents any semantic information of the unverse or object. Encoding means mapping from the unverse to the string or strings of digits. The semantic here is used in the model-theoretic…
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…
Some properties of the optimal representation of numbers are investigated. This representation, which is to the base-e, is examined for coding of integers. An approximate representation without fractions that we call WF is introduced and…
One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph…
This paper contains an algebraic constructive and self-contained account of the invariance rule of the digital root under division for an arbitrary natural basis representation. Both the cases of repeating and non-repeating fractionals are…
This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence…
A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that…
Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels…
In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…
We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…
A phylogenetic tree is a tree with a fixed set of leaves that has no vertices of degree two. In this paper, we axiomatically define four other discrete structures on the set of leaves. We prove that each of these structures is an equivalent…