Abstract. We show the existence of a unital subalgebra Pn of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that Pn is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra Pn contains a two-sided ideal • P n which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form 0 → • P n → Pn → P n−2 → 0. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions n and n 2 +1) by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces Pn and • P n over n ≥ 0 and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

The peak algebra P n is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n − 1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of P n . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of P n and to characterize the elements of P n in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals P j n of P n , j = 0, . . . , n 2 , such that P 0 n is the linear span of sums of permutations with a common set of interior peaks and P n 2 n is the peak algebra. We extend the above results to P j n , generalizing results of Schocker (the case j = 0).

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if (L) is the order complex of a rank (r + 1) geometric lattice L , then for all i ≤ r/2 the h-vector of (L) satisfiesWe also obtain several inequalities for the flag h-vector of (L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of h i−1 ≤ h i when i ≤ 2 7 (r + 5 2 ).

Abstract. We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The resulting allocations of pieces to players are Pareto-optimal with respect to the division. We use a generalization of Sperner's lemma on the polytope of divisions to locate solutions to our generalized cake-cutting problem.

Given a set of p players we consider problems concerning envy-free allocation of collections of k pieces from a given set of goods or chores. We show that if p ≤ n and each player can choose k pieces out of n pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least p 2(k 2 −k+1) players get their desired k pieces each. We further show that if p ≤ k(n − 1) + 1 and each player can choose k pieces, one from each of k cakes that are divided into n pieces each, then there exist a division of the cakes and allocation of the pieces where at least p 2k(k−1) players get their desired k pieces. Finally we prove that if p ≥ k(n− 1)+ 1 and each player can choose one shift in each of k days that are partitioned into n shifts each, then, given that the salaries of the players are fixed, there exist n(1 + ln k) players covering all the shifts, and moreover, if k = 2 then n players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.