Combinatorial proof of the inversion formula on the Kazhdan–Lusztig $$R$$ R -polynomials

Abstract

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\)-polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.