In the modern mathematical literature, Catalan numbers are wonderfully ubiquitous. Although they appear in a variety of disguises, we are so used to having them around, it is perhaps hard to imagine a time when they were either unknown, or known but obscure and underappreciated. It may then come as a surprise that Catalan numbers have a rich history of multiple rediscoveries until relatively recently. Here we review more than 200 years of history, from their first discovery to modern times.
We break the history into short intervals of mathematical activity, each covered in a different section. We spend most of our effort on the early history but do bring it to modern times. We should warn the reader that although this work is in the History of Mathematics, the author is not a mathematical historian. Rather, this work is more of a historical survey with some added speculations based on our extensive reading of the even more extensive literature. Due to the space limitations, this survey is very much incomplete, as we tend to emphasize first discoveries and papers of influence rather than describe subsequent developments.
This paper in part is based on our earlier investigation reported in . Many primary sources are assembled on the Catalan Numbers website , including scans of the original works and their English translations.
The Mongolian astronomer, mathematician, and topographic scientist Minggatu (full name Sharabiin Myangat) (c. 1692–c. 1763), worked at the Qing court in China. Ming's Chinese name is Ming'antu and courtesy name is Jing An. In the 1730s, he wrote a book Quick Methods for Accurate Values of Circle Segments, which included a number of trigonometric identities and power series, some involving Catalan numbers:
He also obtained the recurrence formula
He appears to have no inkling of a combinatorial interpretation of Catalan numbers.