Eulerian number

In <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, the Eulerian number 
 
 
 
 A
 (
 n
 ,
 k
 )
 
 
 {\textstyle A(n,k)}
 
 is the number of <a href="/facts/Permutation/JSw9ukmv">permutations</a> of the numbers 1 to 
 
 
 
 n
 
 
 {\textstyle n}
 
 in which exactly 
 
 
 
 k
 
 
 {\textstyle k}
 
 elements are greater than the previous element (permutations with 
 
 
 
 k
 
 
 {\textstyle k}
 
 "ascents"). 
<a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> investigated them and associated <a href="/facts/Polynomials/Lzak8VVx">polynomials</a> in his 1755 book <a href="/facts/Institutiones_calculi_differentialis/K7kyNEua">Institutiones calculi differentialis</a>.

Other notations for 
 
 
 
 A
 (
 n
 ,
 k
 )
 
 
 {\textstyle A(n,k)}
 
 are 
 
 
 
 E
 (
 n
 ,
 k
 )
 
 
 {\textstyle E(n,k)}
 
 and 
 
 
 
 
 
 ⟨
 
 
 n
 k
 
 
 ⟩
 
 
 
 
 {\displaystyle \textstyle \left\langle {n \atop k}\right\rangle }
 
.

Eulerian number open-in-new

Eulerian number