A set is said to be 3-free if no three elements form an arithmetic progression. Given a 3-free set A of integers 0 = a0 < a1 < · · · < at, the Stanley sequence S(A) = {an} is defined using the greedy algorithm: For each successive n > t, we pick the smallest possible an so that {a0, a1, . . . , an} is 3-free and increasing. Work by Odlyzko and Stanley indicates that Stanley sequences may be div...

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. ...

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisnerand affine monoid algebras. We consider (nonpure) shellable fan’s and the Cohen–Macaulay property. More...