Permutations with extremal number of fixed points

We extend Stanley's 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 excedarices. Several techniques are Used: Desarmenien's desarrangement combinatorics. Gessel's 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. (C) 2008 Elsevier Inc. All rights reserved.