Related papers: Two trees enumerating the positive rationals
Phylogenetic trees play a key role in the reconstruction of evolutionary relationships. Typically, they are derived from aligned sequence data (like DNA, RNA, or proteins) by using optimization criteria like, e.g., maximum parsimony (MP).…
Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients…
The binomial convolution of two sequences $\{a_n\}$ and $\{b_n\}$ is the sequence whose $n$th term is $\sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. If $\{a_n\}$ and $\{b_n\}$ have rational generating functions then so does their binomial…
One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m)…
This paper sets the groundwork for the consideration of families of recursively defined polynomials and rational functions capable of describing the Bernoulli numbers. These families of functions arise from various recursive definitions of…
Knowledge representation and reasoning in law are essential to facilitate the automation of legal analysis and decision-making tasks. In this paper, we propose a new approach based on legal science, specifically legal taxonomy, for…
We consider a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco. We show that extending this Hopf Algebra by identifying pairs of nearest neighbor leaves and producing in this way graphs…
In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing…
Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational…
We formalize the quantum arithmetic, i.e. a relationship between number theory and operator algebras. Namely, it is proved that rational projective varieties are dual to the $C^*$-algebras with real multiplication. Our construction fits all…
We develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi identities. This pairing is an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. In the…
The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and…
Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of…
In this paper we analyse some questions concerning trees on $\kappa$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in…
We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…
We resolve two conjectures of Black-Drellich-Tymoczko about the numbers of valid plane trees for given primary sequences.
A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of…
We provide a new representation of an $\mathbb R$-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the $\mathbb R$-trees with radial and river…
A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…
We give a novel combinatorial interpretation to the perturbative series solutions for a class of Dyson-Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy…