Enumeration of permutations starting with a longest increasing subsequence

We prove a formula for the number of permutations in Sn such that their first n−k entries are increasing and their longest increasing subsequence has length n − k. This formula first appeared as a consequence of character polynomial calculations in the work of Adriano Garsia, [2]. We give an elementary proof of this result and also of its q-analogue. In [2], Adriano Garsia derived as a consequence of character polynomial calculations a simple formula for the enumeration of certain permutations. In his talk at the MIT Combinatorics Seminar [1], he offered a $100 award for an ‘elementary’ proof of this formula. We give such a proof of this formula and its q-analogue. Let Πn,k = {w ∈ Sn|w1 < w2 < . . . wn−k, is(w) = n− k}, the set of all permutations w in Sn, such that their first n− k entries form an increasing sequence and the longest increasing sequence of w has length n− k. The formula in question is the following theorem originally proven by A.Garsia, [2]. Theorem 1. If n ≥ 2k, the number of permutations in Πn,k is given by #Πn,k = k ∑