The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that

     

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h ₄,…), where h ₄ = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h ₂, h ₃, h ₄,…4, 1) are unimodal if h ₄ ≤ 34.     

Distributions of class Lα

L_α (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, L_α is just the classical class L. Several properties for L_α classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some L_α distributions is disproved by giving a counterexample.     

Generalizing the Quinn–Wójs theorem on distinct multiplets of composite Fermions

The combinatorial tool of generating functions for restricted partitions is used to generalize a quantum physics theorem relating distinct multiplets of different angular momenta in the composite Fermion model of the fractional quantum Hall effect. Specifically, if g_ℓ(N,M) denotes the number of distinct multiplets of angular momentum ℓ and total angular momentum M, we prove that

where the sum is taken over all positive divisors of N and L(k)=kℓ-kN/2+3k/2-N+N/(2k)-1/2. The original Quinn–Wójs theorem results when k=1 and it appears that this generalization will be useful in further investigations of nuclear shells modeling elementary particle interactions when the particles are clustered together.     

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function $E(G,x)={\sum }_{i\ge 1}e(G,i)x^i$, where $e(G,i)$ is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph $K_n$, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least $n?3$ non-real roots.     

On the unimodality of the manufacturer's objective function in the newsvendor model

We derive new sufficient conditions for the manufacturer's profit function to be unimodal. We show that if the demand density function is unimodal and right-skewed, then the profit function is quasi-concave over the support of the distribution. We also report some counterintuitive properties of the profit function.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.