Construction of Arbitrary Kazhdan-lusztig Polynomials in Symmetric Groups

To each polynomial P with integral nonnegative coefficients and constant term equal to 1, of degree d, we associate a certain pair of elements (y, w) in the symmetric group Sn, where n = 1 + d + P (1), such that the Kazhdan-Lusztig polynomial Py,w equals P . This pair satisfies l(w)− l(y) = 2d + P (1) − 1, where l(w) denotes the number of inversions of w.