THE NUMBER OF ⊕-SIGN TYPES

Abstract

IN [6], the author introduced the concept of a sign type indexed by the set {(/, j) | 1=£i, j * = n, i ¥=j} for the description of the cells in the affine Weyl group of type A, (in the sense of Kazhdan and Lusztig [3]). Later, the author generalized it by defining a sign type indexed by any root system [8]. Since then, sign type has played a more and more important role in the study of the cells of any irreducible affine Weyl group [1], [2], [5], [8], [9], [10], [12]. Thus it is worth studying sign type itself. One task is to enumerate various kinds of sign types. In [8], the author verified Carter's conjecture on the number of all sign types indexed by any irreducible root system. In the present paper, we consider some special kind of sign types, called ffi-sign types. We show that these sign types are in one-to-one correspondence with the increasing subsets of the related positive root system as a partially ordered set. Thus we get the number of all the ©-sign types indexed by any irreducible positive root system * + by enumerating all the increasing subsets of <J> +. Such a number has a