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Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be theKazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, wepresent a combinatorial proof of the inversion formula on $R$-polynomials dueto Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer'scombinatorial interpretation of the $R$-polynomials in terms of increasingBruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a$V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(\Delta_1,\Delta_2)$of Bruhat paths such that $\Delta_1$ is a decreasing path from $u$ to $w$ and$\Delta_2$ is an increasing path from $w$ to $v$. We find a reflectionprinciple on $V$-paths, which leads to a combinatorial proof of the inversionformula. Moreover, we give two applications of the reflection principle. First,we restrict this involution to $V$-paths from $u$ to $v$ with maximal length.This provides a direct interpretation for the equi-distribution property thatany nontrivial interval $[u,v]$ has as many elements of even length as elementsof odd length. This property was obtained by Verma in his derivation of theM\"obius function of the Bruhat order. Second, using the reflection principlefor the symmetric group, we obtain a refinement of the inversion formula byrestricting the summation to permutations ending with a given element.

Combinatorial Proof of the Inversion Formula on the Kazhdan-Lusztig R-Polynomials