I'm drafting an OEIS sequence, and I've formulated a few conjectures. I was hoping someone here could help me to prove them.
Definition
Let the descent set of a permutation $\omega \in S_n$ be the set $$ D(\omega) = \{i : \omega(i) > \omega(i + 1)\}. $$
Let $$ d(\omega) = \begin{cases} 0 & \omega = \operatorname{id} \\ \min(D(\omega)) & \text{else} \end{cases}. $$
And let $$ a(n) = \sum_{\omega \in S_n} d(\omega). $$
Example
For example, if the permutation (written as a word) is $$ \omega = 4\ 3\ 7\ 1\ 6\ 5\ 2 $$
Then the descent set is $D(\omega) = \{1, 3, 5, 6\}$ because $$ \begin{align*} \omega(1) = 4 &> \omega(2) = 3, \\ \omega(3) = 7 &> \omega(4) = 1, \\ \omega(5) = 6 &> \omega(6) = 5, \text{ and}\\ \omega(6) = 5 &> \omega(7) = 2, \end{align*} $$ and the minimal element of the descent set is $d(\omega) = \min\{1, 3, 5, 6\} = 1$.
The sequence of $a(n)$ starting at $n=1$ begins
0, 1, 7, 37, 201, 1231, 8653, 69273, 623521, 6235291, 68588301, 823059733, 10699776673, 149796873591, 2246953104061, 35951249665201, 611171244308673, 11001082397556403, 209020565553571981, 4180411311071439981, 87788637532500240001, 1931350025715005280463
Question
I have proven that $$ a(n) = \sum_{k=1}^{n-1} k^2\binom{n}{k+1}(n - k - 1)!, $$ and I have two conjectures that I need help in proving (or disproving):
- $a(n+1) = (n+1)a(n) + n^2$, with $a(1) = 0$.
- $\displaystyle\lim_{n\rightarrow\infty} \frac{a(n)}{n!} = e - 1.$
$d(\omega)$ is overcomplicated. It's simply the length of the strictly ascending prefix in $\omega$. However your exception of $\omega = \text{id}$ has further obfuscated things. When $\omega = \text{id}$ you should have $d(\omega) = n$ to be consistent with this observation.
When you fix your definition like this you will find this OEIS sequence already exists: A002627. Your sequence only differs by $a(n) = \text{A002627}(n) - n$, so I don't think it warrants another entry. All your observations are also already present in the comments.