3 Sigma Sequences
For a chromosome \(X\), the sigma sequence is the function \(\sigma _X(k) = \mathrm{sig}(X^{(k)}) \in \mathbb {Q} \times \mathbb {Q}\). We write \(a_X(k) = \sigma _X(k).1\) and \(b_X(k) = \sigma _X(k).2\).
The sigma sequence is antitone (componentwise decreasing).
There exists \(K\) such that \(\sigma _X(k) = 0\) for all \(k \ge K\).
The components of the sigma sequence satisfy interlacing inequalities. If \(k\) is even, then \(b_X(k+1) \le a_X(k)\) and \(a_X(k+1) \le b_X(k)\). If \(k\) is odd, the inequalities are reversed.
If \(X \in \mathrm{Pi}\), then \(a_X(k) - b_X(k+1) \le 1\) and \(b_X(k) - a_X(k+1) \le 1\).
If \(X {\lt} Y\) in \(\mathrm{Pi}\), then \(\sigma _X(k) \le \sigma _Y(k)\) componentwise for all \(k\) with strict inequality for some \(k\).