Given $n$ cards, labelled $1$ to $n$, what is the expectation of the sum of the last decreasing sequence?
e.g.
$n=4$
Then {$1,2,3,4$} gives $0$,
whilst {$2,1,3,4$} gives $2+1=3$
and {$3,4,1,2$} gives $4+1=5$
I tried using recursion and inserting the new $(n+1)^{th}$ card at different points of old sequences, but it didn't work.