CM fields with a reciprocal unit-primitive element

Let K be a noncyclotomic CM field. We show that the field $K\cap \mathbb{R}$ has a reciprocal unit-primitive element when K does. Also, we prove some related conditions that make the converse of this assertion true.     

Counting derangements, involutions and unimodal elements in the wreath product Cr ? Sn

We count derangements, involutions and unimodal elements in the wreath product C _r ≀ S _n by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group S _n and on the hyperoctahedral group B _n when r = 1, 2, respectively.     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

On multiple pattern avoiding set partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T.