 Eulerian number

With the descent convention, the Eulerian number $A(n,k)$ counts permutations of $1,\dots ,n$ with exactly $k$ positions $i$ satisfying $a_i>a_{i+1}$. These positions are permutation descents. For $n=3$, the row is $(1,4,1)$: only 123 has no descent and only 321 has two. Inserting the largest label giveswith $A(0,0)=1$ and out-of-range entries zero. Of the $n$ insertion gaps, $k+1$ preserve $k$ descents, while the remaining gaps increase the descent count. Some sources shift the second index by one, so the convention must accompany the symbol.^[ref]

Eulerian numbers, denoted as \( E(n, k) \), are a set of integers that count the number of permutations of \( n \) elements in which exactly \( k \) elements appear in ascents. An ascent in a permutation is a position where the next element is larger than the current one.