The Sequence Catalan Didn't Discover

Take a convex polygon with seven sides and ask how many ways you can cut it into triangles using non-crossing diagonals. The answer is 42. Take a hexagon: 14. A pentagon: 5. A square: 2. The sequence 1, 1, 2, 5, 14, 42, 132, 429 turns up everywhere combinatorial objects nest inside themselves — balanced parentheses, binary trees, ways a stack can sort a list. C_10 is 16,796.

Leonhard Euler hit it on September 4, 1751, when he wrote to Christian Goldbach about counting polygon triangulations. He worked the cases up to a ten-sided polygon by hand, took ratios of consecutive terms, and guessed the closed form. Goldbach wrote back in October with a way to verify it. By December Euler had a proof using the binomial theorem.

Except it had already been done. In 1730, in Beijing, an astronomer at the Qing imperial court named Mingantu — born in Inner Mongolia around 1692, full name Sharavyn Myangat — started writing a treatise called Ge Yuan Mi Lu Jie Fa, "The Quick Method for Obtaining the Precise Ratio of Division of a Circle." He was working out infinite-series expansions of sin(2α) and sin(4α) in terms of sin(α), the kind of formula Jesuit missionaries had brought into China. The coefficients he kept getting back were the Catalan numbers, used recursively, decades before Euler's letter.

Mingantu died around 1763 with the manuscript unfinished. His student Chen Jixin completed it in 1774. Nobody published it for another sixty years.

The sequence picked up Eugène Catalan's name in the 19th century after he linked it to a problem about parenthesizing products. Mingantu's role wasn't recognized in the Western literature until 1988. The numbers have an older name in Chinese mathematics now — Ming Antu numbers — that almost no one outside the field uses.