Related papers: Binary trees using the bookshelf and baseball cons…
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this…
We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. The price to pay for this…
In this series of articles we study connections between combinatorics of multidimensional generalizations of Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we…
In this paper, we study tree--like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of…
We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e a…
We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct…
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…
An extension of order theory is presented that serves as a formalism for the study of dendroidal sets analogously to way the formalism of order theory is used in the study of simplicial sets.
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are new.
Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a $C^*$-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in…
We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum…
We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure…
A few notes about infinite trees in a descriptive set-theoretic setting.
We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains…
Let $G$ be a connected graph. The Jacobian group (also known as the Picard group or sandpile group) of $G$ is a finite abelian group whose cardinality equals the number of spanning trees of $G$. The Jacobian group admits a canonical simply…
We describe arithmetic computations in terms of operations on some well known free algebras (S1S, S2S and ordered rooted binary trees) while emphasizing the common structure present in all them when seen as isomorphic with the set of…
As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin-Ma-Ma-Zhou in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we…