In combinatorics, the (Pascal) Binomial Triangle more or less started the entwining of combinatorics, probability and algebra.
The other two most seminal and ubiquituous triangles of numbers are the Stirling (duals between 1st and 2nd sort, permutation world and set world) and the Eulerian numbers (permutations seen as words on ordered alphabet).
For links between combinatorics and number theory, I think the first prize would be the Bernouilli numbers.