The purpose of this paper is to identify, as far as possible, those sequences in the Encyclopedia of Integer Sequences which count orbits of an infinite permutation group acting on n-sets or n-tuples of elements of the permutation domain. The paper also provides an introduction to the properties of such sequences and their relations with combinatorial enumeration problems.

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric fu...

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon, etc. In some cases enumeration gives new expression for sequences enumerating these structures.

We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible applications to Interaction-Round-a-Face and hard-particle statistical models defined on random lattices.

A square involution is a square permutation which is also an involution. In this paper we give the enumeration of square involutions, using purely combinatorial methods, by establishing a bijective correspondence with a class of lattice paths. As a corollary to our result, we enumerate various subclasses of square involutions, including the classes of triangular, decomposable, and fat involutions.

Combinatorial auctions – in which bidders can bid on combinations of goods – can increase the economic efficiency of a trade when goods have complementarities. Recent theoretical developments have lessened the computational complexity of these auctions, but the issue of cognitive complexity remains an unexplored barrier for the online marketplace. This study uses a data-driven approach to explo...

Combinatorial design theory traces its origins to statistical theory of experimental design but also to recreational mathematics of the 19th century and to geometry. In the past forty years combinatorial design theory has developed into a vibrant branch of combinatorics with its own aims, methods and problems. It has found substantial applications in other branches of combinatorics, in graph th...

Enumerative combinatorics is about counting. The typical question is to find the number of objects with a given set of properties. However, enumerative combinatorics is not just about counting. In “real life”, when we talk about counting, we imagine lining up a set of objects and counting them off: 1, 2, 3, . . .. However, families of combinatorial objects do not come to us in a natural linear ...

The notion of succession rule (system for short) provides a powerful tool for the enumeration of many classes of combinatorial objects. Often, different systems exist for a given class of combinatorial objects, and a number of problems arise naturally. An important one is the equivalence problem between two different systems. In this paper, we show how to solve this problem in the case of syste...

An interesting combinatorial (enumeration) problem arises in the initial phase of the polyhedral homotopy continuation method for computing all solutions of a polynomial equation system in complex variables. It is formulated as a problem of finding all solutions of a specially structured system of linear inequalities with a certain additional combinatorial condition. This paper presents a compu...