# How to grow it? Strategies of mathematical development presented by the example of enumerating certain set partitions

Mathematik in der Lehre

## Abstract

We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are the number of ways of partitioning a finite set into a fixed number $$d$$ of subsets of odd or even size, into $$a$$ parts of odd and $$b$$ parts of even size, and into $$d$$ parts, each of which has a size in a certain congruence class modulo some natural number $$m$$. To this end, pattern guessing, recursion and induction, combinatorial interpretation and generating functions are employed. The participants of the dialogue represent different perspectives on and approaches to mathematics.

## Keywords

Mathematics education Heuristics Philosophy of mathematics Combinatorics Generating functions Set partitions

## MSC-Classification

00A35 97D20 05A19 68R15 97K20 05A18

## References

1. 1.
de Mier, A.: Lecture notes for ‘Enumerative Combinatorics’. University of Oxford (2004). https://math.dartmouth.edu/archive/m68f07/public_html/lectec.pdf. Accessed: 21 Nov 2019Google Scholar
2. 2.
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete mathematics, 2nd edn. Addison-Wesley Publishing Company, New York (1994)
3. 3.
Lakatos, I.: Proofs and refutations. In: Worrall, J., Zahar, E. (eds.) The logic of mathematical discovery. Cambridge University Press, Cambridge (1976)
4. 4.
Polya, G.: Induction and analogy in mathematics. Mathematics and plausible reasoning, vol. 1. Birkhaeuser, Basel (1988)
5. 5.
Schneps, L.: Book review (Grothendieck-serre correspondence). The Mathematical Intelligencer 30, 60 (2008).