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 flipclasses, we introduce some 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.