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The peak algebra and the descent algebras of types B and D
We show the existence of a unital subalgebra β n 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 β n is
Discrete Envy-free Division of Necklaces and Maps
We study the discrete variation of the classical cake-cutting problem where n players divide a 1-dimensional cake with exactly (n-1) cuts, replacing the continuous, infinitely divisible "cake" with a
Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices
It is proved that the order complex of a geometric lattice has a convex ear decomposition and a combinatorial flag h-vector proof of hi-1 ≤ hi when i ≤ (2/7)(r + (5/2)).
The Peak Algebra of the Symmetric Group
The peak set of a permutation σ is the set {i : σ(i − 1) < σ(i) > σ(i + 1)}. The group algebra of the symmetric group Sn admits a subalgebra in which elements are sums of permutations with a common
Fair division with multiple pieces
Annihilators of permutation modules
Permutation modules are fundamental in the representation theory of symmetric groups $\Sym_n$ and their corresponding Iwahori--Hecke algebras $\He = \He(\Sym_n)$. We find an explicit combinatorial
A Hierarchy of Classes of Bounded Bitolerance Orders
A more comprehensive list of classes of bitolerance orders is provided and equality between some of these classes in general and other classes in the bipartite domain is proved.
Linear Inequalities for Rank 3 Geometric Lattices
A collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices are given.
New results on the peak algebra
The peak algebra $$\mathfrak{P}_{n}$$ 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