Important numbers in Combinatorics

Question

I recently went through some important numbers like the Stirling and Bell number for calculation of partitions /equivalence relations.

I was wondering if someone can help me get a list of important numbers and their applications in Combinatorics ; like Catalan , Fibonacci , Stirling etc.???

Answer 1

Here are some that I’ve encountered; the Wikipedia links generally give some indication of their applications.

Bell numbers
ordered Bell (or Fubini) numbers
Catalan numbers
Delannoy numbers
Euler(ian) numbers
Fibonacci numbers
Genocchi numbers
Lah numbers
Motzkin numbers
partition numbers
Schröder numbers
Schröder-Hipparchus numbers
Stirling numbers of the first kind
Stirling numbers of the second kind

And perhaps we should include a couple of non-integer sequences:

Bernoulli numbers
Harmonic numbers

Added: And we also have the nimbers or Grundy numbers, though they are the ordinal numbers with new operations of addition and multiplication rather than a distinguished subset of the integers.

Answer 2

There are some sequences which I feel embarrassed that I do not recognize immediately (well, I do know the Catalan numbers). Here are these numbers sequences (or arrays of numbers sometimes) which I feel I encounter on a daily basis.