Related papers: Growth rates for subclasses of Av(321)

We study $321$-avoiding affine permutations, and prove a formula for their enumeration with respect to the inversion number by using a combinatorial approach. This is done in two different ways, both related to Viennot's theory of heaps.…

Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or…

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank…

We consider a class of stochastic growth models on the integer lattice which includes various interesting examples such as the number of open paths in oriented percolation and the binary contact path process. Under some mild assumptions, we…

We bound the number of permutations with a fixed number $r$ of $321 \ominus p_0$ patterns by a constant times the number of permutations which avoid $321 \ominus p_0$. We use this new upper bound to show that the ordinary generating…

This work provides a structural characterisation of hereditary graph classes that do not contain a star forest, several graphs obtained from star forests by subset complementation, a union of cliques, and the complement of a union of…

We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a $2$-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an…

We propose a class of evolutionary models that involves an arbitrary exchangeable process as the breeding process and different selection schemes. In those models, a new genome is born according to the breeding process, and then a genome is…

In this paper we describe an approach to construct large extendable collections of vectors in predefined spaces of given dimensions. These collections are useful for neural network latent space configuration and training. For classification…

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known…

Preservation theorems provide a direct correspondence between the syntactic structure of first-order sentences and the closure properties of their respective classes of models. A line of work has explored preservation theorems relativised…

Since network motifs are an important property of networks and some networks have the behaviors of rewiring or reducing or adding edges between old vertices before new vertices entering the networks, we construct our non-randomized model…

A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of…

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its asymptotic behaviour is an interesting…

We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$…

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7…

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…

Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the…

The width of a neural network matters since increasing the width will necessarily increase the model capacity. However, the performance of a network does not improve linearly with the width and soon gets saturated. In this case, we argue…