ar X iv : m at h / 06 09 18 4 v 1 [ m at h . C O ] 6 S ep 2 00 6 FACES OF GENERALIZED PERMUTOHEDRA

ar X iv : m at h / 06 09 18 4 v 2 [ m at h . C O ] 1 8 M ay 2 00 7 FACES OF GENERALIZED PERMUTOHEDRA

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f -, hand γ-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving descent statistics. This includes a combinatorial interpret...

ar X iv : m at h / 06 09 42 6 v 3 [ m at h . C O ] 1 5 O ct 2 00 6 SUM - PRODUCT ESTIMATES IN FINITE FIELDS VIA KLOOSTERMAN

We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.

ar X iv : h ep - p h / 06 09 07 7 v 1 7 S ep 2 00 6 Hadron multiplicities in e + e − annihilation with heavy primary quarks

ar X iv : m at h / 06 09 42 6 v 2 [ m at h . C O ] 5 O ct 2 00 6 SUM - PRODUCT ESTIMATES IN FINITE FIELDS VIA KLOOSTERMAN

We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.

ar X iv : m at h / 06 09 13 5 v 1 [ m at h . C O ] 5 S ep 2 00 6 Score sets in oriented bipartite graphs

The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture when | A | = 1, 2, 3, or when A is a ge...