Combinatorics essentially means systematic counting.

Some simple and widely applicable parts of Combinatorics are as follows. Permutation, combinations, selections, partitions, generating series, pigeonhole principles, inclusion–exclusion principles. Group-theory-based enumerations, graphs, posets,and lattices.

Applications include to Computer Science, Chemistry, Geometry and the Feynman diagrams of Quantum Physics. Combinatorics is also useful for gaining a basic understanding of some topics in Topology: notions of connectivity, of traversability, of cover and of separation, topological spaces, and knots.

So far, we have systematically introduced useful relative difference variables
and ratio variables are systematically introduced for combinatorial objects.

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 the totality of each kind of object. We have also given a systematic account of arenas for the basic combinatorial objects. Power sets and the Young lattice of partitions are two previously already well-known examples.

Arenas are 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 and thus a type of Combinatorics – or differential and metric Geometry.

See here

for a text developing this point of view. It is a book, covering around 3 semesters’ worth of material. The first 4 parts – basic Combinatorics and Group Theory – are suitable for first or second year undergraduates. The next 3 parts – Graph Theory and Order Theory – are suitable for second or third year undergraduates. And Part VIII supplies this material with largely new research on this material.

Contact us if interested in Combinatorics materials, at a level equivalent to second through to fifth year of university courses. Quite a lot of which is new research that can none the less be understood people with this amount of experience experience.

Overall, we are choosing to interpret Combinatorics from the extended point of view of not just sequences of counts of mathematical structures but also of more sequences of graphs formed by, and orders on, these mathematical structures. The above link is also our first installment as regards building an Encyclopaedia of Graphs and Orders. Focusing in particular on those that arise in basic and widely applicable mathematics. Expect a large number of graphs and orders arising from basic Geometry to join this material over 2024 and 2025.