Permutations with extremal number of fixed points

We extend Stanleyʹs work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Désarménienʹs desarrangement combinatorics, Gesselʹs hook-factorization and the analytical properties of two new permutation statistics “DEZ” and “lec.” Explicit formulas for the maximal case are derived by using symmetric function tools.