Bijective enumeration of permutations starting with a longest increasing subsequence

TL;DR

This paper derives a formula for counting permutations with specific initial increasing sequences and longest increasing subsequence length, providing elementary bijective proofs including a RSK-based and a permutation-only approach.

Contribution

It introduces two elementary bijective proofs for a formula on permutations with constrained LIS and initial increasing segments, including a new q-analogue.

Findings

Derived a formula for permutations with specified initial increasing segments and LIS length

Provided two elementary bijective proofs, one using RSK and one permutation-only

Extended the result to a q-analogue

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.