  Regular semisimple Hessenberg varieties are smooth subvarieties of the flag
variety, and their examples contain the flag variety itself and the
permutohedral variety which is a toric variety. We give a complete
classification of Fano and weak Fano regular semisimple Hessenberg varieties in
type A in terms of combinatorics of Hessenberg functions. In particular, we
show that if the anti-canonical bundle of a regular semisimple Hessenberg
variety is nef, then it is in fact nef and big.
