Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials

In 1979, Kazhdan and Lusztig defined a family Pu,v(q) of polynomials indexed by pairs of elements of a Coxeter group W that have proven to be fundamental objects of study in representation theory. At the same time, they can be defined combinatorially, and so have also been studied by combinatorialists. Although it is now known that Pu,v(q) depends only on the structure of the Bruhat interval [u, v] as an abstract poset, explicit formulas which exhibit this invariance are only known in general for intervals of length at most 4. In this paper we use a formula of Brenti to give an explicit formula for the quadratic coefficient of Pu,v(q) which is almost combinatorially invariant, and use this formula to give a combinatorially invariant formula for intervals of length at most 6 in the special case that u = e.