Maeckes logo

<    1    >


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   → 2     2 1 3     2 3 1     3 1 2
A(3, 2) = 1   → 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!.

 


Additional information

The Swiss mathematician Leonhard Euler described this triangle in 1755.

 


Deutsch   Español   Français   Nederlands