Geodesic
Bijective enumeration of permutations starting with a longest increasing subsequence
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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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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     author = {Panova, Greta},
     title = {Bijective enumeration of permutations starting with a longest increasing subsequence},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2829},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2829/}
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Panova, Greta. Bijective enumeration of permutations starting with a longest increasing subsequence. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2829. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2829/

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