Combinatorics

Special numbers

Catalan numbers, derangements and Stirling numbers.

For checking your work after you’ve tried it yourself.

Step 1:
Choose a kind

Catalan, derangements, Stirling or Bell.

Step 2:
Type n

And k where it's needed.

Step 3:
Read the value

With its neighbours, and every object for small n.
What it does.
Gives Catalan numbers, derangements, Stirling numbers of both kinds and Bell numbers, with nearby values and, for small nn, a list of everything being counted.
When it helps.
Checking a value you got from a recurrence, or seeing what the numbers actually count.
What it won’t do.
n up to 500, or 200 for Stirling and Bell numbers.
Example.
D4=9D_4 = 9: there are 9 ways to rearrange 4 letters so that none stays put.

CnC_{n}: ways to write n pairs of brackets that match (also mountain paths, triangulations of a polygon, binary trees).

Exact, however many digits.

Try