How to grow it? Strategies of mathematical development presented by the example of enumerating certain set partitions
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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.
KeywordsMathematics education Heuristics Philosophy of mathematics Combinatorics Generating functions Set partitions
MSC-Classification00A35 97D20 05A19 68R15 97K20 05A18
- 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