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.
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Notes
The controversy between ‘tricky’ and ‘systematic’ mathematics has a long history. See e.g. [5], p. 64, on the correspondence between Serre and Grothendieck: “Serre was the more open-minded of the two; any proof of a good theorem, whatever the style, was liable to enchant him, whereas obtaining even good results ‘the wrong way’ – using clever tricks to get around deep theoretical obstacles – could infuriate Grothendieck.”
This notation for the set of all positive integers up to \(m\) will be used throughout.
The book is [2]. It provides a good overview of identities for the Stirling numbers of the second kind.
THE ON-LINE ENCYCLOPEDIA OF INTEGER SEQUENCES: https://oeis.org.
See e.g. [1], p. 47.
References
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Probably, most of the combinatorial results developed here are already known. For instance, the generating functions for the number of partitions of a finite set into a fixed number of odd or even blocks are mentioned in [1]. As the emphasis of this article lies mainly on reflecting different ways of developing mathematics we do not discuss the issue of our results’ novelty in detail.
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Carl, M., Schmitz, M. How to grow it? Strategies of mathematical development presented by the example of enumerating certain set partitions. Math Semesterber 67, 237–261 (2020). https://doi.org/10.1007/s00591-019-00267-y
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DOI: https://doi.org/10.1007/s00591-019-00267-y
Keywords
- Mathematics education
- Heuristics
- Philosophy of mathematics
- Combinatorics
- Generating functions
- Set partitions