Two classes of posets with real-rooted chain polynomials

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets with this property, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to irreducible finite Coxeter groups, are presented here. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set $\{1, 2,\dots,n\}$ which have ascents at specified positions is shown to be real-rooted, hence unimodal, and a good estimate for the location of the peak is deduced.Comment: 19 page

Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen-Macaulay posets. As a result, new families of convex polytopes whose barycentric subdivisions have real-rooted $f$-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag $h$-vector, convex polytope, barycentric subdivisio

QoS-based elasticity for service chains in distributed edge cloud environments

With the emerging IoT and Cloud-based networked systems that rely heavily on virtualization technologies, elasticity becomes a dominant system engineering attribute for providing QoS-aware services to their users. Although the concept of elasticity can introduce significant QoS and cost benefits, its implementation in real systems is full of challenges. Indeed, nowadays systems are mainly distributed, built upon several layers of abstraction, and with centralized control mechanisms. In such a complex environment, controlling elasticity in a centralized manner might strongly penalize scalability. To overcome this issue, we can conveniently split the system in autonomous subsystems that implement elasticity mechanisms and run control policies in a decentralized manner. To efficiently and effectively cooperate with each other, the subsystems need to communicate among themselves to determine elasticity decisions that collectively improve the overall system performance. This new architecture calls for the development of new mechanisms and efficient policies. In this chapter, we focus on elasticity in IoT and Cloud-based systems, which can be geo-distributed also at the edge of the networks, and discuss its engineering perspectives along with various coordination mechanisms. We focus on the design choices that may affect the elasticity properties and provide an overview of some decentralized design patterns related to the coordination of elasticity decisions