In combinatorics, the [(Pascal) Binomial Triangle][1] more or less started the entwining of combinatorics, probability and algebra.

The other two most seminal and ubiquituous triangles of numbers are the [Stirling][2] (duals between 1st and 2nd sort, permutation world and set world, the most often generalized family of numbers) and the [Eulerian][3] numbers (permutations seen as words on ordered alphabet), the reference statistics on permutations.

For links between combinatorics and number theory, I think the first prize would be the [Bernoulli][4] numbers.


  [1]: http://en.wikipedia.org/wiki/Pascal's_triangle
  [2]: http://en.wikipedia.org/wiki/Stirling_number
  [3]: http://en.wikipedia.org/wiki/Eulerian_number
  [4]: http://en.wikipedia.org/wiki/Bernoulli_number