Let W be an irreducible finite or affine Coxeter group and let W c be the set of fully commutative elements in W. We prove that the set W c is closed under the Kazhdan-Lusztig preorder LR if and only if W c is a union of two-sided cells of W .

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n 1 , . . . , n m , n m+1 = n 1 , and 0 ≤ j ≤ m − 1,which generalizes a result of Calkin [Acta Arith. 86 (1998), 17-26]. Moreover, we show that for all positive integers n, r and j,where A = (r − 1) n 2 + r j+1 2 + k 2 − rjk, which solves a problem raised by Zudilin [Electron. J. Combin. 11 (2004), #R22].

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein * over Z 2. We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the h-polynomial of the complex coincides with the polynomial counting twisted involutions by descents. In particular, this gives a type-independent proof that the latter is symmetric.