A short proof of a sum of powers formula

Therefore, we need to count how many k + 1 tuples (a1, a2, . . . , ak, i) exist with the constraints 1 ≤ am ≤ i ≤ n for all 1 ≤ m ≤ k. To simplify, make the change of variable = i+ 1 and count the tuples (a1, a2, . . . , ak, ) such that 1 ≤ am < ≤ n+ 1. Let j ∈ {1, . . . , k}. Consider a subset S of {1, 2, . . . , n+ 1} with j + 1 elements. We want to count the number of tuples (a1, a2, . . . , ak, ) satisfying 1 ≤ am < ≤ n+ 1 and that the values taken by the elements in the tuple are exactly those in S. Since > am for all m, we see that is forced to be the maximum among the elements of S. Then we have the other j possible values from S distributed among a1, a2, . . . , ak. There are {k j } ways of partitioning {a1, a2, . . . , ak} into j blocks.1 Then there are j! ways of assigning values to those blocks from the values left in S. Therefore we have j! {k j } ways of matching the tuples to the values of S. There are (n+1 j+1 )