Permutation patterns, Stanley symmetric functions, and the Edelman-Greene correspondence

Abstract

Generalizing the notion of a vexillary permutation, we introduce a filtration of $S_{\infty}$ by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by avoiding a finite set of patterns. In doing so, we show that if $w$ is a permutation containing $v$ as a pattern, then there is an injection from the set of Edelman-Greene tableaux of $v$ to the set of Edelman-Greene tableaux of $w$ which respects inclusion of shapes. We also consider the set of permutations whose Edelman-Greene tableaux have distinct shapes, and show that it is closed under taking patterns.

Keywords

• edelman-greene correspondence
• stanley symmetric functions
• specht modules
• pattern avoidance
• [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]