Bus network design is an important problem in public transportation. A main step to this design is determining the number of required terminals and their locations. This is a special type of facility location problem, which is a time-consuming, large scale, combinatorial problem. In a previous attempt by the authors, this problem had been solved by GAMS, based on a branch and bound algorithm.&...

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck’s generalized Ferrers diagrams, Wright’s stacks, and Andrews’ convex compositions. These results describe combinatorial properties, generating functions, and asymptotic f...

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 five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2...

Bus network design is an important problem in public transportation. A main step to this design is determining the number of required terminals and their locations. This is a special type of facility location problem, which is a time-consuming, large scale, combinatorial problem. In a previous attempt by the authors, this problem had been solved by GAMS, based on a branch and bound algorithm. ...

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partition...

With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to highlight the correspondence with other combinatorial objects with the same cardinality. Then we will study the structures obtained by ordering properly independen...

Until now, the enumeration of connected graphs has been dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enum...