Abstract. In this paper, we solve an open problem on distributive lattices, which was proposed by Stanley in 1998. This problem was motivated by a conjecture due to Griggs, which equivalently states that the incomparability graph of the boolean algebra [Formula: see text] is nice. Stanley introduced the idea of studying the nice property of a graph by investigating the Schur positivity of its corresponding chromatic symmetric function. Since the boolean algebras form a special class of distributive lattices, Stanley raised the question of whether the incomparability graph of any distributive lattice is Schur positive. Stanley further noted that this seems quite unlikely. We construct a family of distributive lattices which are not nice and hence not Schur positive. We also provide a family of distributive lattices which are nice but not Schur positive.

Motivated by the concepts of the inverse Kazhdan--Lusztig polynomial and the equivariant Kazhdan--Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan--Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan--Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan--Lusztig polynomials, and we determine the equivariant inverse Kazhdan--Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot, and Young's formula for the equivariant Kazhdan--Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr, and Radcliffe's combinatorial interpretation for the ordinary Kazhdan--Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan--Lusztig polynomials.

We prove that, for any positive $q$, the $q$-Eulerian polynomial of type $D$ has only real zeros. This settles an open problem of Brenti in 1994. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which was originally conjectured by Brenti and recently proved by Savage and Visontai.