What is known for sure (and is mentioned in all the books I've seen so far) is that for homonuclear diatomics the "$\pi \rightarrow \sigma$ switch" happens when going from $\ce{N2}$ to $\ce{O2}$: for 14-electron case $\pi_{2p_{x,y}}$ orbitals are lower in energy then $\sigma_{2p_{z}}$, for 16-electon one it is other way around. The usual "explanation" is that moving from $\ce{Li2}$ to $\ce{F2}$ the successive decrease of the $s-p$ interaction between the $2s$ orbital of one atom and the $2p_z$ orbital of another atom decreases the energy of the $\sigma_{2p_{z}}$. But you have to understand that this is merely a manifestation of a known fact: we know how orbitals of these molecules are arranged from the molecular orbital theory calculations and use the notion of $s-p$ interaction to rationalize this knowledge.
The problem is that no information about 15-electron case can be inferred from the above mentioned knowledge about molecular orbitals of homonuclear diatomics from $\ce{Li2}$ to $\ce{F2}$. A priori we could not tell will $\sigma_{2p_{z}}$ orbital be lower in energy than $\pi_{2p_{x,y}}$ orbitals or higher then them for a 15-electron case. Besides, things coul be different for different 15-electron cases, such as $\ce{N2-}$, $\ce{O2+}$, $\ce{NO}$, etc.: for some $\sigma_{2p_{z}}$ orbital might be lower than $\pi_{2p_{x,y}}$ orbitals, for others - higher. There is no simple rule to use, one have to resort to molecular orbital theory calculations to tell the truth.
