Eulerian numbers

Abstract

Leonhard Euler certainly didn’t have permutations on his mind when in 1755 he wrote about the numbers bearing his name in this chapter. (He was trying to find an easy way to compute sums of the form $1^k + 2^k + 3^k +\cdots$ for any value of k.) Nowadays, however, these numbers are usually defined combinatorially: in terms of permutations.

“If you don’t make mistakes, you’re not working on hard enough problems. And that’s a big mistake.”

–Frank Wilczek