Introduction

There exists a large gap between the empirical evidence of the computational capabilities of neural networks and our ability to systematically analyze and design those networks. Although it is well known that classical Fourier analysis is a very effective mathematical tool for the design and analysis of linear systems, such a tool was not available for artificial neural networks, which are inherently nonlinear. In the late 1980s, the spectral analysis tool was introduced in the domain of discrete neural networks. The application of the spectral technique led to a number of new insights and results, including lower and upper bounds on the complexity of computing with neural networks as well as methods for constructing optimal (in terms of performance) feedforward networks for computing various arithmetic functions.

The focus of the presentation in this chapter is on an elementary description of the basic techniques of Fourier analysis and its applications in threshold circuit complexity. Our hope is that this chapter will serve as background material for those who are interested in learning more about the progress and results in this area. We also provide extensive bibliographic notes that can serve as pointers to a number of research results related to spectral techniques and threshold circuit complexity.

Abstract

Permuting machines were the early inspiration of the theory of permutation pattern classes. Some examples are given which lead up to distilling the key properties that link them to pattern classes. It is shown how relatively simple ways of combining permuting machines can lead to quite complex behaviour and that the notion of regularity can sometimes be used to contain this complexity. Machines which are sensitive to their input data values are shown to be connected to a more general notion than pattern classes. Finally some open problems are presented.

Introduction

Although permutation patterns have only recently been studied in a systematic manner their history can be traced back many decades. It could be argued that the well-known lemma of Erdős and Szekeres is really a result about pattern classes (a pattern class whose basis contains both an increasing and a decreasing permutation is necessarily finite). However it is perhaps more convincing to attribute the birth of the subject to the ground-breaking first volume of Donald Knuth's Art of Computer Programming series. In the main body of his text, and in some fascinating follow-up exercises, Knuth enumerated some pattern classes, and found some bases, while at the same time introducing some techniques on generating functions that, in due time, were codified as “the kernel method”.

Abstract

The paper offers an overview over selected results in the literature on partially ordered patterns (POPs) in permutations, words and compositions. The POPs give rise in connection with co-unimodal patterns, peaks and valleys in permutations, Horse permutations, Catalan, Narayana, and Pell numbers, bi-colored set partitions, and other combinatorial objects.

Introduction

An occurrence of a pattern τ in a permutation π is defined as a subsequence in π (of the same length as τ) whose letters are in the same relative order as those in τ. For example, the permutation 31425 has three occurrences of the pattern 1-2-3, namely the subsequences 345, 145, and 125. Generalized permutation patterns (GPs) being introduced in allow the requirement that some adjacent letters in a pattern must also be adjacent in the permutation. We indicate this requirement by removing a dash in the corresponding place. Say, if pattern 2-31 occurs in a permutation π, then the letters in π that correspond to 3 and 1 are adjacent. For example, the permutation 516423 has only one occurrence of the pattern 2-31, namely the subword 564, whereas the pattern 2-3-1 occurs, in addition, in the subwords 562 and 563. Placing “[” on the left (resp., “]” on the right) next to a pattern p means the requirement that p must begin (resp., end) from the leftmost (resp., rightmost) letter.

Abstract

Structural methods as applied to the study of classical permutation pattern avoidance are introduced and described. These methods allow for more detailed study of pattern classes, answering questions beyond basic enumeration. Additionally, they frequently can be applied wholesale, producing results valid for a wide collection of pattern classes, rather than simply ad hoc application to individual classes.

Introduction

In the study of permutation patterns, the important aspects of permutations of [n] = {1, 2, …, n} are considered to be the relative order of both the argument and the value. Specifically, we study a partial order, denoted ≼ and called involvement, on the set of such permutations where π ∈ Sk is involved in σ ∈ Sn, i.e. π ≼ σ if, for some increasing function f : [k] → [n] and all 1 ≤ i < j ≤ k, σ(i) < σ(j) if and only if π(f(i)) < π(f(j)). This dry and uninformative definition is necessary to get us started, but the reader should certainly be aware that another definition of involvement is that some of the points in the graph of π can be erased so that what remains is the graph of σ (possibly with a non-uniform scale on both axes) – in other words the pattern of σ (its graph) occurs as part of the pattern of π.

Abstract

A descent in a permutation α1α2 · αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic which was first studied by P. A. MacMahon almost a hundred years ago, and it still plays an important role in the study of permutations. Representing set partitions by equivalent canonical sequences of integers, we study this statistic among the set partitions, as well as the numbers of rises and levels. We enumerate set partitions with respect to these statistics by means of generating functions, and present some combinatorial proofs. Applications are obtained to new combinatorial results and previously-known ones.

Introduction

A descent in a permutation α = α1α2 ··· αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic. This statistic was first studied by MacMahon, and it still plays an important role in the study of permutation statistics. In this paper we study the statistics of numbers of rises, levels and descents among set partitions expressed as canonical sequences, defined below.

Abstract

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance–or the prescribed number of occurrences–of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.

Introduction

Patterns in permutations have been studied sporadically, often implicitly, for over a century, but in the last two decades this area has grown explosively, with several hundred published papers. As seems to be the case with most things in enumerative combinatorics, some instances of permutation patterns can be found already in MacMahon's classical book from 1915, Combinatory Analysis. In the seminal paper Restricted permutations of Simion and Schmidt from 1985 the systematic study of permutation patterns was launched, and it now seems clear that this field will continue growing for a long time to come, due to its plethora of problems that range from the easy to the seemingly impossible, with a rich middle ground of challenging but solvable problems.

Abstract

This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises four topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs and 4. ultimate modular periodicity of numbers of MSOL-definable structures.

Introduction

We survey some general results in combinatorial enumeration. A problem in enumeration is (associated with) an infinite sequence P = (S1, S2, …) of finite sets Si. Its counting function fP is given by fP (n) = |Sn|, the cardinality of the set Sn. We are interested in results of the following kind on general classes of problems and their counting functions.

Scheme of general results in combinatorial enumeration. The counting function fP of every problem P in the class C belongs to the class of functions F. Formally, {fP | P ∈ C} ⊂ F.

The larger C is, and the more specific the functions in F are, the stronger the result. The present overview is a collection of many examples of this scheme.

A very brief introduction to permutation patterns

We say a permutation π contains or involves the permutation σ if deleting some of the entries of π gives a permutation that is order isomorphic to σ, and we write σ ≤ π. For example, 534162 contains 321 (delete the values 4, 6, and 2). A permutation avoids a permutation if it does not contain it.

This notion of containment defines a partial order on the set of all finite permutations, and the downsets of this order are called permutation classes. For a set of permutations B define Av(B) to be the set of permutations that avoid all of the permutations in B. Clearly Av(B) is a permutation class for every set B, and conversely, every permutation class can be expressed in the form Av(B).

For the problems we need one more bit of notation. Given permutations π and σ of lengths m and n, respectively, their direct sum, π ⊕ σ, is the permutation of length m + n in which the first m entries are equal to π and the last n entries are order isomorphic to σ while their skew sum, π ⊖ σ, is the permutation of length m + n in which the first m entries are order isomorphic to π while the last n entries are equal to π.