Related papers: An easy upper bound for Ramsey numbers
A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…
The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor…
We developed computer algebra tools for enumerating conjugacy classes of independent subsets and generating sets of symmetric groups up to $n=7$, and carried out an initial analysis of the obtained results.
We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Partition functions for non-interacting particles are known to be symmetric functions. It is shown that powerful group-theoretical techniques can be used not only to derive these relationships, but also to significantly simplify calculation…
In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…
Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures.…
We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…
We continue the investigation, that began in [3] and [4], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm…
Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if $G$ is an…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
We introduce the notion of asymptotic partition regularity for Diophantine equations. We show how this notion is at the core of almost all known negative results in the Ramsey theory of equations, and we use it to produce new ones, as in…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
Residual finiteness growth gives an invariant that indicates how well-approximated a finitely generated group is by its finite quotients. We briefly survey the state of the subject. We then improve on the best known upper and lower bounds…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of…
Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $\alpha_{G,n}$ denote the number of simultaneous conjugacy…
In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in…
The rational homology group of the order complex of non-even partitions of a finite set is calculated. A twisted version of the Goresky-MacPherson approach to similar homology calculations is proposed.