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We extend Stanley's work on alternating permutations with extremal number offixed points in two directions: first, alternating permutations are replaced bypermutations with a prescribed descent set; second, instead of simply countingpermutations we study their generating polynomials by number of excedances.Several techniques are used: Desarmenien's desarrangement combinatorics,Gessel's hook-factorization and the analytical properties of two newpermutation statistics "DEZ" and "lec". Explicit formulas for the maximal caseare derived by using symmetric function tools.

Permutations with Extremal number of Fixed Points