Bijective enumeration of permutations starting with a longest increasing subsequence

Details Bijective enumeration of permutations starting with a longest increasing subsequence Bijective enumeration of permutations starting with a longest increasing subsequence

2009-05-13

arxiv.org/abs/0905.2013v2

Mathematics

Combinatorics

Revised version: added a bijection via permutations only, without RSK; incorporated referee's suggestions. To appear in DMTCS

Scientific paper

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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