We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the -genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework.
Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant such that every family of subsets of an n-element set that contains no odd-sunflower consists of at most sets. We construct such families of size at least . We also characterize minimal odd-sunflowers of triples.
It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR property. Given a prime power , we study Peisert-type graphs of order without the strict-EKR property and with the minimum number of edges and we call such graphs extremal. We determine number of edges in extremal graphs for each value of q. If q is a square or a cube, we show the uniqueness of the extremal graph and classify all maximum cliques explicitly. Moreover, when q is a square, we prove that there is no Hilton-Milner type result for the extremal graph, and show the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph.
Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety with and any n-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes . We apply combinatorial methods and we give concrete formulas for these volumes.
We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the -representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.
We study the normal Cayley graphs on the symmetric group , where and is the set of all cycles in with length in I. We prove that the strictly second largest eigenvalue of can only be achieved by at most four irreducible representations of , and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when I contains neither nor n we know exactly when has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of , and we obtain that does not have the Aldous property whenever . As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of where .
In a recent paper (arXiv:2301.09744), Erickson and Hunziker consider partitions in which the arm–leg difference is an arbitrary constant m. In previous works, these partitions are called -asymmetric partitions. Regarding these partitions and their conjugates as highest weights, they prove an identity yielding an infinite family of dimension equalities between and modules. Their proof proceeds by the manipulations of the hook content formula. We give a simple combinatorial proof of their result.
Let be the set of all mappings . The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each is chosen uniformly at random (i.e., with probability ). The cycle of T contained within its largest component is called the deepest one. For any , let denote the length of this cycle. In this paper, we establish the convergence in distribution of and find the limits of its expectation and variance as . For n large enough, we also show that nearly 55% of all cyclic vertices of a random mapping lie in its deepest cycle and that a vertex from the longest cycle of T does not belong to its largest component with approximate probability 0.075.
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