Christoffel Words and the Calkin-Wilf Tree
نویسندگان
چکیده
In this note we present some results on the Calkin-Wilf tree of irreducible fractions, giving an insight on the duality relating it to the Stern-Brocot tree, and proving noncommutative versions of known results relating labels of the CalkinWilf trees to hyperbinary expansions of positive integers. The main tool is the Christoffel tree introduced in a paper by Berstel and de Luca.
منابع مشابه
Sturmian words and the Stern sequence
Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of characteristic Sturmian words. A natural arithmetization of the theory is obtained by representing central and Christoffel words by irreducible fractions labeling respectively two binary trees, the Raney (or Calkin-Wilf) tree and the Stern-Brocot tree. ...
متن کاملSTATISTICAL PROPERTIES OF THE CALKIN–WILF TREE: REAL AN p−ADIC DISTRIBUTION
We examine statistical properties of the Calkin–Wilf tree and give number-theoretical applications. 1. A mean-value related to the Calkin–Wilf tree The Calkin–Wilf tree is generated by the iteration a b 7→ a a + b , a+ b b , starting from the root 1 1 ; the number a a+b is called the left child of a b and a+b b the right child; we also say that a b is the mother of its children. Recently, Calki...
متن کاملThe q-Calkin-Wilf tree
We define a q-analogue of the Calkin–Wilf tree and the Calkin–Wilf sequence. We show that the nth term f (n; q) of the q-analogue of the Calkin–Wilf sequence is the generating function for the number of hyperbinary expansions of n according to the number of powers that are used exactly twice. We also present formulae for branches within the q-analogue of the Calkin–Wilf tree and predecessors an...
متن کاملLinking the Calkin-Wilf and Stern-Brocot trees
Links between the Calkin-Wilif tree and the Stern-Brocot tree are discussed answering the questions: What is the jth vertex in the nth level of the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the hyperb...
متن کاملAn arborist’s guide to the rationals
There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as ‘transpose shadows’ of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 18 شماره
صفحات -
تاریخ انتشار 2011