Eulerian number

In combinatorics, the Eulerian number is the number of permutations of the numbers 1 to in which exactly elements are greater than the previous element (permutations with "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis. Other notations for are and . The Eulerian polynomials are defined by the exponential generating function The Eulerian numbers may be defined as the coefficients of the Eulerian polynomials: An explicit formula for is For fixed there is a single permutation which has 0 ascents: . Indeed, as for all , . This formally includes the empty collection of numbers, . And so . For the explicit formula implies , a sequence in that reads . Fully reversing a permutation with ascents creates another permutation in which there are ascents. Therefore . So there is also a single permutation which has ascents, namely the rising permutation . So also equals . A lavish upper bound is . Between the bounds just discussed, the value exceeds . For , the values are formally zero, meaning many sums over can be written with an upper index only up to . It also means that the polynomials are really of degree for . A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle. Values of for are: {| class="wikitable" style="text-align:right;" |- ! ! width="50" | 0 ! width="50" | 1 ! width="50" | 2 ! width="50" | 3 ! width="50" | 4 ! width="50" | 5 ! width="50" | 6 ! width="50" | 7 ! width="50" | 8 |- ! 0 | 1 || || || || || || || || |- ! 1 | 1 || || || || || || || || |- ! 2 | 1 || 1 || || || || || || || |- ! 3 | 1 || 4 || 1 || || || || || || |- ! 4 | 1 || 11 || 11 || 1 || || || || || |- ! 5 | 1 || 26 || 66 || 26 || 1 || || || || |- ! 6 | 1 || 57 || 302 || 302 || 57 || 1 || || || |- ! 7 | 1 || 120 || 1191 || 2416 || 1191 || 120 || 1 || || |- ! 8 | 1 || 247 || 4293 || 15619 || 15619 || 4293 || 247 || 1 || |- ! 9 | 1 || 502 || 14608 || 88234 || 156190 || 88234 || 14608 || 502 || 1 |} For larger values of , can also be calculated using the recursive formula This formula can be motivated from the combinatorial definition and thus serves as a natural starting point for the theory.