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

  • Merlin Carl
  • Michael SchmitzEmail author
Mathematik in der Lehre


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.


Mathematics education Heuristics Philosophy of mathematics Combinatorics Generating functions Set partitions 


00A35 97D20 05A19 68R15 97K20 05A18 


  1. 1.
    de Mier, A.: Lecture notes for ‘Enumerative Combinatorics’. University of Oxford (2004). 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)zbMATHGoogle Scholar
  3. 3.
    Lakatos, I.: Proofs and refutations. In: Worrall, J., Zahar, E. (eds.) The logic of mathematical discovery. Cambridge University Press, Cambridge (1976)zbMATHGoogle Scholar
  4. 4.
    Polya, G.: Induction and analogy in mathematics. Mathematics and plausible reasoning, vol. 1. Birkhaeuser, Basel (1988)zbMATHGoogle Scholar
  5. 5.
    Schneps, L.: Book review (Grothendieck-serre correspondence). The Mathematical Intelligencer 30, 60 (2008). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Abteilung für MathematikEuropa-Universität FlensburgFlensburgGermany

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