I was recently looking at Eulerian numbers for a problem and I noticed a nice, and probably lesser/never known, property on their recursive definition. In summary, it seemed like an exercise problem out of a Combinatorics textbook except I couldn't find one.

Eulerian numbers $A(n,k)$ count the number of permutation of numbers from $1$ to $n$ in which exactly $k$ elements are smaller than the previous element. In other, words they count the number of permutation with exactly $k$ descents.

With this interpretation, we can setup the following recurrence as shown in this Proposition 3 of this lecture.

$A(n,k)=(n-k)A(n-1,k-1)+(k+1)A(n-1,k)$

The first term here corresponds to the number of permutations in which the element $n$ contributes to the descent count and the second term to those where it doesn't.

The problem that I was interested in was that of counting the number of permutation with $k$ descents and end with a descent.

After some thought, it seemed easy to setup a recurrence for this count as well.

Let $A_d(n,k)$ be the number of permutations of $n$ elements with exactly $k$ descents and end with a descent. Let $A_a(n,k)$ be the same count but those that end with an ascent. Then,

$A_d(n,k)=k A_d(n-1,k)+A_a(n-1,k-1)+(n-k)A_d(n-1,k-1)$

The first term here corresponds to inserting $n$ in an existing descent. The second term is where we insert $n$ as second to last position thereby simultaneously making the permutation end with an descent and increasing the descent count.

The last one is where we take a permutation with $n-1$ elements. There are '$n-1$' gaps where we can insert $n$ (we don't $n$ to be in the last position as it would make the permutation to end with an ascent). Of the $n-1$ gaps, $k-1$ are already descents which means there are $n-k$ gaps that correspond to an ascent. Inserting $n$ here would increase descent count by one giving us the requisite permutation.

Similarly, we can write the following recurrence for the $A_a(n-k)$

$A_a(n,k)=A_d(n-1,k)+(n-k-1)A_a(n-1,k-1)+(k+1)A_a(n-1,k)$

Using the fact that $A(n,k)=A_a(n,k)+A_d(n,k)$ and some simple manipulations, we get

$A_d(n,k)=(n-k)A(n-1,k-1)$ and $A_a(n,k)=(k+1)A(n-1,k)$

I hope you can see now why this surprised me. The recurrence usually written for Eulerian numbers also has another simpler combinatorial representation without any change!!

That is, the number of permutations with $k$ descents in which $n$ contributes to the descent count is equinumerous with the permutations with $k$ descents that end with a descent.

Hope you enjoyed this. See ya soon.

Until then, Yours Aye

Me