Flipclasses and Combinatorial Invariance for Kazhdan–Lusztig polynomials

In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials for the symmetric group. Using the new concept of flip classes, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of q(h )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.