A formula for certain inverse Kazhdan–Lusztig polynomials in Sn

Let w0 denote the permutation [n,n−1,…,2,1]. We give a new explicit formula for the Kazhdan–Lusztig polynomials Pw0w,w0x in Sn when x indexes an irreducible component of the singular locus of the Schubert variety Xw. To do this, we utilize a standard identity that relates Px,w and Pw0w,w0x.