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Bijective enumeration of permutations starting with a longest increasing subsequence

Abstract : We prove a formula for the number of permutations in $S_n$ 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 recent work of Adriano Garsia and Alain Goupil. We give two "elementary' bijective proofs of this result and of its q-analogue, one proof using the RSK correspondence and one only permutations.
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Greta Panova. Bijective enumeration of permutations starting with a longest increasing subsequence. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.973-982. ⟨hal-01186257⟩

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