Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials

In this work, we investigate a new approach to the Combinatorial Invariance Conjecture of Kazhdan--Lusztig polynomials for the symmetric group. We introduce some new combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of qʰ of the Kazhdan--Lusztig R-polynomials, for h≤ 6. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.