How do you determine shorthand notation?

But given the section where you posted your question, I guess you mean shorthand with respect to the electron configuration of atoms. And so I will address shorthand notation in this context.......

Now given that chemical reactions occur on the basis of sharing and transfer of electrons, we need only to represent the valence shell of electrons; the non-valence shells, the inner core electrons, are along for the ride, and the chemical action only occurs for the valence electrons.......

And so, for say iron, #Z=26#, we could write the longhand notation.....

#1s^(2)2s^(2)2p^(6)3s^(2)3p^(6)4s^(2)3d^(6)#;

But the #1s^(2)2s^(2)2p^(6)3s^(2)3p^(6)# #"inner core"# is precisely that of the LAST Nobel gas....#Z=18#, i.e. ARGON. And so to save us a bit of unnecessary work, we could write.......

#[Ar]4s^(2)3d^(6)#; where #[Ar]# specifies the #1s^(2)2s^(2)2p^(6)3s^(2)3p^(6)# configuration.........

And for #Fe^(2+)#, and #Fe^(3+)#, I could write......

#{[Ar]3d^(6)}^(2+)#, and #{[Ar]3d^(5)}^(3+)# respectively........

If I have missed the bus, and this was not relevant to your question, I apologize.........