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Euler's triangle
The Euler triangle shows the development of Eulerian numbers.
Explanation
The Eulerian numbers A(n, m) give the number of permutations of the integers 1 to n in which exactly m elements are greater than the previous element. You see it at
A(3, 0) = 1 → 3 2 1 A(3, 1) = 4 → 1 3 2 2 1 3 2 3 1 3 1 2 A(3, 2) = 1 → 1 2 3
In the table are the values of the Eulerian numbers for n = 1 to 9 and m = 0 to 8.
n m 0 1 2 3 4 5 6 7 8 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
The sum of the n-th line is the total of all permutations, so the factorial n!.
HistoryThe Swiss mathematician Leonhard Euler described this triangle in 1755. |