Pattern Recognition on Oriented Matroids

4Farey Subsequences

The Farey sequence n of order n is defined to be the increasing sequence of irreducible fractions $\frac{h}{k}\in \mathbb{Q}suchthat\frac{0}{1}\le \frac{h}{k}\le \frac{1}{1}$ and k ≤ n. For example,

$F_6=\left(\frac{0}{1}<\frac{1}{6}<\frac{1}{5}<\frac{1}{4}<\frac{1}{3}<\frac{2}{5}<\frac{1}{2}<\frac{3}{5}<\frac{2}{3}<\frac{3}{4}<\frac{4}{5}<\frac{5}{6}<\frac{1}{1}\right).$

Let C be a finite set of cardinality n := |C| > 1, and A its proper subset; m := |A|. The increasing sequence of fractions

$\begin{array}{c}T\left(\mathbb{B}\left(n\right),m\right):=\left(\frac{\left|B\cap A\right|}{\gcd \left(\left|B\cap A\right|,\left|B\right|\right)}/\frac{\left|B\right|}{\gcd \left(\left|B\cap A\right|,\left|B\right|\right)}:B\subseteq C,\text{}\text{}\left|B\right|>0\right)\\ =\left(\frac{h}{k}\in T_n:m+k-n\le h\le m\right),\end{array}\left(4.1\right)$

where (n) as earlier denotes the Boolean ...