Geodesic
Orphans in forests of linear fractional transformations
Sandie Han1 ; Ariane M. Masuda1 ; Satyanand Singh1 ; Johann Thiel1
1New York City College of Technology
The electronic journal of combinatorics, Tome 23 (2016) no. 3
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A positive linear fractional transformation (PLFT) is a function of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c$ and $d$ are nonnegative integers with determinant $ad-bc\neq 0$. Nathanson generalized the notion of the Calkin-Wilf tree to PLFTs and used it to partition the set of PLFTs into an infinite forest of rooted trees. The roots of these PLFT Calkin-Wilf trees are called orphans. In this paper, we provide a combinatorial formula for the number of orphans with fixed determinant $D$. In addition, we derive a method for determining the orphan ancestor of a given PLFT. Lastly, taking $z$ to be a complex number, we show that every positive complex number has finitely many ancestors in the forest of complex $(u,v)$-Calkin-Wilf trees.
DOI : 10.37236/5684
Classification : 05A18, 05C05
Mots-clés : continued fractions, Calkin-Wilf tree, linear fractional transformations

Sandie Han  1   ; Ariane M. Masuda  1   ; Satyanand Singh  1   ; Johann Thiel  1

1 New York City College of Technology 
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Sandie Han; Ariane M. Masuda; Satyanand Singh; Johann Thiel. Orphans in forests of linear fractional transformations. The electronic journal of combinatorics, Tome 23 (2016) no. 3. doi: 10.37236/5684

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