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Leonhard Euler's 1751 correspondence already studied triangulations of a convex polygon and gave a closed formula for their counts. Johann Andreas von Segner supplied a combinatorial recurrence relation in a paper written in 1758. Eugène Charles Catalan's nineteenth-century work connected the formula to bracketed products. This history, documented in Igor Pak's account, shows how the same counting structure reappeared through different objects; the modern name alone does not establish discovery priority.^[ref]

By Igor Pak.

Online text: arxiv.org/pdf/1408.5711

Standalone historical preprint, arXiv:1408.5711. Version 1 was submitted on 25 August 2014; the inspected version 2 is dated 27 August 2014. Its cover says the account will appear as an appendix in a forthcoming book by Richard P. Stanley; the authorship here applies to this preprint, not to the whole book.

Cited passages: §§2–3 and §5, printed pp. 2–4: Euler correspondence, Segner recurrence and Catalan's bracketed products.

Igor Pak works in discrete mathematics, including enumerative combinatorics, algebraic combinatorics, polyhedral geometry, and probability and computation on groups. His research page presents these as interconnected subjects and lists lectures on convex polytopes and combinatorial inequalities.^[ref] A common theme is to explain a numerical counting identity through the structure of the objects being counted, such as a bijection, rather than relying only on algebraic manipulation. Polyhedral methods add geometric constraints to those discrete enumeration questions.

Homepage: www.math.ucla.edu/~pak/