Step 1: Understanding the Concept:
Bond order = \((N_b - N_a)/2\).
Step 2: Detailed Explanation:
O\(_2^{2-}\): BO = 1, O\(_2^-\): BO = 1.5, O\(_2\): BO = 2, O\(_2^+\): BO = 2.5
Increasing order: 1, 1.5, 2, 2.5
Step 3: Final Answer:
O\(_2^{2-}\), O\(_2^-\), O\(_2\), O\(_2^+\).