The Eulerian number 〈n k〉 gives the number of permutations of {1, 2, ..., n} having k permutation ascents . Note that a slightly different definition of Eulerian number is used by Comtet, who defines the Eulerian number A(n, k) (sometimes also denoted A_(n, k)) as the number of permutation runs of length k - 1, and hence A(n, k) = 〈n k - 1〉. The Eulerian numbers are given explicitly by the sum 〈n k〉 = sum_(j = 0)^(k + 1) (-1)^j(n + 1 j)(k - j + 1)^n (Comtet 1974, p. 243).

combination lock | Euler number | Euler prime | Euler's number triangle | Euler zigzag number | permutation ascent | permutation run | polylogarithm | Simon Newcomb's problem | sinc function | Worpitzky's identity | Z-transform