Related papers: Generating Bijections between HOAS and the Natural…
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on…
This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our…
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally…
The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and…
We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call…
Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for all $n$, precisely when $p$ and $q$ have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit…
In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…
The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian…
We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only…
Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the…
In this paper, we give part-preserving bijections between three fundamental families of objects that serve as natural framework for many problems in enumerative combinatorics. Specifically, we consider compositions, Dyck paths, and…
When using a third language to construct a bilingual dictionary, it is necessary to discriminate equivalencies from inappropriate words derived as a result of ambiguity in the third language. We propose a method to treat this by utilizing…
This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics…
Starting with an inclusion-exclusion proof of a combinatorial identity, a direct bijection can be produced using recursive subtraction (sometimes with a direct combinatorial description). We apply this method to identities for generalized…
We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…
We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is…
These notes present an approach to obtaining monoid operations which are compatible with a given family of mappings in the sense that the mappings become left translations in the monoid. This can be applied to various situations such as the…
In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.
Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of…
It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed,…