Applied Combinatorics

Back cover description

Material covered: Combinatorics essentially means systematic counting. We present some Combinatorics topics that are both simple and widely applicable, by which they are very much worth knowing,
understanding and applying. These include permutations, combinations, other selections, partitions, generating series, and pigeonhole and inclusion-exclusion principles. Group-theory-based enumerations, and especially graphs, posets, and lattices. Applications covered include to Computer Science, Chemistry, Geometry and to the Feynman diagrams of Quantum Physics. And toward starting to gain a basic understanding of some topics in Topology: notions of connectivity, of traversability, of cover, and of separation,
topological spaces, and a brief introduction to knots.
This is a Mathematics book, and yet we start from
scratch as regards Readers interested in applications
who are not (primarily) Mathematicians. A total of almost 1000 exercises, worked proofs and worked examples
are included.

Some of our innovations: Useful relative difference variables and ratio variables are systematically introduced, and applied toward understanding not only the graphs but also the spaces of graphs. And not only the partitions but also the spaces of partitions, and so on.
These developments are applying successful ‘Shape Theory’ or ‘relational’ ideas – usually found in Geometry
and in high-end approaches to Physics – now to basic Combinatorics as well. In particular,
the theory of each type of mathematical object
involves not only the structures of those objects
but also the structure of the arena: the space formed by
those objects. This is one of the ways in which
Topology enters the theory of every other STEM subject,
often accompanied by one of Order Theory – posets
and lattices – or Geometry. These parts of the current Volume are not only presented accessibly for beginners,
but are also innovative enough to source new research in Mathematics and in many other areas of STEM.
Throughout, Conceptual Thinking, including finding Truer Names for all the main objects and results encountered, is included and encouraged.

Our format: This Volume has been made freely available
in electronic form in support of the new Generation Z’s
considerable desire to become self-sufficient. Two affordably-priced series of Books are growing from
the current Volume. Since these will on many occasions
refer to the current Volume, its free availability can effectively be viewed as part of this affordable pricing.