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We prove a formula for the number of permutations in $S_n$ such that theirfirst $n-k$ entries are increasing and their longest increasing subsequence haslength $n-k$. This formula first appeared as a consequence of characterpolynomial calculations in recent work of Adriano Garsia and Alain Goupil. Wegive two `elementary' bijective proofs of this result and of its $q$-analogue,one proof using the RSK correspondence and one only permutations.

Bijective enumeration of permutations starting with a longest increasing subsequence