Orphans in Forests of Linear Fractional Transformations
نویسندگان
چکیده
A positive linear fractional transformation (PLFT) is a function of the form f(z) = az+b cz+d where a, b, c and d are nonnegative integers with determinant ad− bc 6= 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.
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عنوان ژورنال:
- Electr. J. Comb.
دوره 23 شماره
صفحات -
تاریخ انتشار 2016