Invariant Principal Order Ideals under Foata's Transformation

Let Φ denote Foata’s second fundamental transformation on permutations. For a permutation σ in the symmetric group Sn, let Λ̃σ = {π ∈ Sn : π 6w σ} be the principal order ideal generated by σ in the weak order 6w. Björner and Wachs have shown that Λ̃σ is invariant under Φ if and only if σ is a 132-avoiding permutation. In this paper, we consider the invariance property of Φ on the principal order ideals Λσ = {π ∈ Sn : π 6 σ} with respect to the Bruhat order 6. We obtain a characterization of permutations σ such that Λσ are invariant under Φ. We also consider the invariant principal order ideals with respect to the Bruhat order under Han’s bijection H. We find that Λσ is invariant under the bijection H if and only if it is invariant under the transformation Φ.