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The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P. The h∗-polynomial of P is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the h∗-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the h∗-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the h∗-polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for h∗-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all h∗-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

Enumeration of derangements in the symmetric group 𝔖 n is classical. Extensions of the enumerative results to the hyperoctahedral group Bn are combinatorially sound. That in the even-signed permutation group Dn remains largely unexplored. Let d n D ( q ) = ∞ σ ∈ D n D q m a j ( σ ) be the generating function of derangements in Dn by their major indices. We study in this work properties of d n D ( q ) , including recurrence relations and factorial generating function. By proving the ratio monotonicity of d n D ( q ) , the unimodality, log-concavity and spiral property of d n D ( q ) are also established.

We answer a nonnegativity problem of G. E. Andrews and D. Newman by the q-unimodality of Gaussian polynomials. Some new considerations of the q-log-concavity and q-unimodality of Gaussian polynomials from a purely partition-theoretic perspective will also be presented.

We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.

In this paper, we study unimodality conditions for distributions that describe markets with stochastic demand. Such conditions naturally emerge in the analysis of game-theoretic models of market competition (Cournot games) and supply chain coordination (Stackelberg games). We express the price elasticity of expected demand in terms of the mean residual life (MRL) function of the demand distribution and characterize optimal prices or equivalently, points of unitary elasticity, as fixed points of the MRL function. This leads to economic interpretable conditions on the demand distribution under which such fixed points exist and are unique. We find that markets with increasing price elasticity of expected demand that eventually become elastic correspond to distributions with decreasing generalized mean residual life (DGMRL) and finite second moment. DGMRL distributions strictly generalize the widely used increasing generalized failure rate (IGFR) distributions. We further elaborate on the relationship of the two classes, link their limiting behavior at infinity and examine moment and closure properties of DGMRL distributions that are important in economic applications. The DGMRL unimodality condition is useful in the analysis of optimal decisions under uncertainty in settings that are not covered by the widely-used IGFR condition; thus, it can be of broader interest to the game-theory and operations research literature.

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

The use of decision trees for obtaining and representing clustering solutions is advantageous, due to their interpretability property. We propose a method called Decision Trees for Axis Unimodal Clustering (DTAUC), which constructs unsupervised binary decision trees for clustering by exploiting the concept of unimodality. Unimodality is a key property indicating the grouping behavior of data around a single density mode. Our approach is based on the notion of an axis unimodal cluster: a cluster where all features are unimodal, i.e., the set of values of each feature is unimodal as decided by a unimodality test. The proposed method follows the typical top-down splitting paradigm for building axis-aligned decision trees and aims to partition the initial dataset into axis unimodal clusters by applying thresholding on multimodal features. To determine the decision rule at each node, we propose a criterion that combines unimodality and separation. The method automatically terminates when all clusters are axis unimodal. Unlike typical decision tree methods, DTAUC does not require user-defined hyperparameters, such as maximum tree depth or the minimum number of points per leaf, except for the significance level of the unimodality test. Comparative experimental results on various synthetic and real datasets indicate the effectiveness of our method.

A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G . In this paper we introduce the general position polynomial as ∑ i ≥ 0 a i x i , where a i is the number of distinct general position sets of G with cardinality i . The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K ( n , 2), with unimodal general position polynomials are presented.

The stabilization of solutions by distributed feedback control functions for second- and third-order ordinary differential equations (ODEs) has been presented in earlier studies. The present paper extends these results to the stabilization of n-th order ODEs using a distributed control function expressed in integral form with first-order derivatives. The problem of stabilizing n-th order ODE solutions by distributed control functions is significantly more complex and nontrivial. This work introduces a method for selecting the parameter set within the distributed control function. Furthermore, the connection between palindromic polynomials, log-concavity, and stability with respect to initial conditions (Lyapunov stability) in n-th order ODEs with distributed feedback control functions is established. We use the symmetry property of palindromic polynomials.

In this work we introduce a wide family of continuous univariate distributions with support ( 0 , ∞ ) that includes as special cases the majority of classical continuous distributions. The new family contains distributions with cumulative distribution function of the form F ( x ; θ ) = g - 1 ( h ( x ; θ ) ) , where g and h satisfy specific conditions. We study its properties, including aging, tail properties and unimodality, and apply our general results to families of classical distributions, thereof obtaining alternative proofs of well known results. We also discuss how the new framework can be exploited for the generation of new distributions that possess specific desirable properties (e.g. they have heavy tails, monotone failure rates etc).