Related papers: A proof of the Stanley--Stembridge conjecture

We compute the stable homology of the braid group with coefficients in any Schur functor applied to the integral reduced Burau representation. This may be considered as a hyperelliptic analogue of the Mumford conjecture (Madsen--Weiss…

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…

We investigate chromatic symmetric functions in the relation to the algebra $\Gamma$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $\Gamma$ in terms of chromatic symmetric functions. We also consider…

We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of "typical" covering maps, and use graph Galois theory to "symmetrize" the conjecture. The conjecture is then…

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a…

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, ie…

Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and…

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context.…

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…

We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply…

This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…

We consider a linear relation which expresses Stanley's chromatic symmetric function for a poset in terms of the chromatic symmetric functions of some closely related posets, which we call the modular law. By applying this in the context of…

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space…

We investigate the distributions of the number of: (1) monochromatic complete subgraphs over edgewise 2-colorings of complete graphs; and (2) monochromatic arithmetic progressions over 2-colorings of intervals, as statistical Ramsey theory…

We investigate the possibility of proving upper bounds on Hadwiger's number of a graph with partial information, mirroring several known upper bounds for the chromatic number. For each such bound we determine whether the corresponding bound…

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for…

For time-reversal invariant graphs we prove the Bohigas-Giannoni-Schmit conjecture in its most general form: For graphs that are mixing in the classical limit, all spectral correlation functions coincide with those of the Gaussian…

Following the lead of Stanley and Gessel, we consider a morphism which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing…