Catalan Numbers Day - May 14

Happy Catalan Numbers Day!

Catalan numbers are a sequence of numbers that represent the counts of several mathematical concepts. They're named after Eugene Charles Catalan, a French-Belgian mathematician who worked on number theory and combinatorics. 

One of the things Catalan numbers count are the number of Dyck words, which are words consisting of X's and Y's which:

• Consist of an equal number of X's and Y's,
• And where all beginning sub-words have their number of X's greater than or equal to their number of Y's.

For example, XYXY is a Dyck word because XYX, XY, and X are all beginning sub-words where there are at least as many X's as Y's. But XYYX is not a Dyck word because XYY is a beginning sub-word where there are more Y's than X's. Note that if you substitute "(" for X and ")" for Y, Dyck words constitute a valid sequence for arithmetic or regular expressions.

Here are the smallest Dyck words.

This video provides more uses of Catalan numbers.

The Catalan numbers can be calculated using the formula:

• C(n) = (2n)!/(n!n!(n+1))

Though you're no doubt anxious to calculate them yourself, I've taken the liberty to calculate the smallest (and easiest) ones for you:

We celebrate Catalan numbers today because 5 and 14 are the largest Catalan numbers that form a (US-style) date: May 14.