#catalan numbers

[dailyhistoryposts]

Minggatu

A famous Mongolian astronomer and mathematician, Minggatu (1692-1763) was a scientist at the Qing Court.

Born what is now Inner Mongolia of the Sharaid clan, he was an excellent student. He worked at the Imperial Observatory on drafting the calendar and studying the armillary sphere (a model of objects in the sky).

Of note, he was the first person in China to calculate infinite series and he obtained more than 10 formulae. He also first established what would later be called Catalan numbers, a sequence of natural numbers helpful in dealing with recursively define objects. Though these numbers would be named after a European, it was Minggatu who discovered them. His work is incredibly remarkable in that he worked with series, trigonometry, and logarithms with algebra and without differential and integral calculus.

Also of note was his work in surveying the Dhungar Khanate, which allowed the first atlas of China drawn with scientific methods, the Complete Atlas of the Empire, to be finished.

For three years, he was the administrator of the Imperial Astronomical Bureau. His courtesy name was Jing An. He was an innovator in math, science, and analytical researcher. In 2002, a minor planet was named after him.

[useless-catalanfacts]

After a long time of looking like

thanks to @useless-nigeriafacts I know what these are!

So apparently these numbers are part of a sequence called Catalan numbers, named after the Belgian mathematician Eugène Charles Catalan (you had a cool surname dude)

So… thanks anon for letting me know

[thepinkairplane]

On the first night of Hanukkah, we light one candle. on the second night we light two candles. on the third night we light five candles. on the fourth night we light fourteen candles. on the fifth night we... hm. wait a minute

[eri-pl]

Ok so I reminded myself that Catalan numbers exist, so now for yet another thing nobody asked for: mapping the 14 rooted trees with 4 nodes (+root) to the 14 Valar because why wouldn't I?

[ibemused]

Solutions to 3rd, 4th, (and even 5th) degree polynomials created using truncation of (algebraic) power series to derive approximate solutions. By Professor Norman Wildberger from University of New South Wales, who "rejects" irrational (e.g., pi, e, i), and radical (e.g., roots) numbers

The new array of numbers: "Geodes" "extend these Catalan numbers from a one-dimensional to multi-dimensional array based on the number of ways a polygon can be divided using non-intersecting lines"

BTW, the nth Catalan number (Cn) represents the number of ways to triangulate a polygon with n+2 sides. Catalan numbers count recursive objects

Mathematician solves algebra's oldest problem using intriguing new number sequences

A UNSW Sydney mathematician has discovered a new method to tackle algebra's oldest challenge—solving higher polynomial equations.

[ebookspksite]

Free download Catalan Numbers written by Richard P. Stanley .

[isomorphismes]

Catalan numbers for

triangulations

gambler's ruin

binary trees

trees

…from the Flajolet/Sedgewick coursera analysis of algorithms

in R:

catalan <- function(n) { if(n<=1) 1 else lapply(1:n, FUN=function(k) catalan(k-1) * catalan(n-k) ) %>% unlist %>% sum }

(This is slow above catalan(10) though, so here's a memoised version. Then if you want to run above catalan(250) you'll need to adjust options(expressions=5e3) to something higher; otherwise the nesting of calls goes too deep.)

[forklady42]

The Catalan sequence is probably the most frequently encountered sequence that is still obscure enough to cause mathematicians lacking access to Sloane’s A Handbook of Integer Sequences to expend inordinate amounts of energy rediscovering formulas that were worked out long ago.

Martin Gardner