Let $P_n$ be the partitions set of $[n]$ . Let $f: P_n \to N_0$ be a statistic that suits for each block/class, the number of the block/class that contains the maximum number $n$. Write a recurrence relation for calculating the exponential generating function of the partitions number in $P_n$ according to the statistic f.
For $P_3$ for example I found that the partitions and the values of $f$ are:
{1}/{2}/{3} $f=3$
{12}/{3} $f=2$
{13}/{2} $f=1$
{1}/{23} $f=2$
{123} $f=1$
I tried hard to find a relation between the partitions $P_n$ and the statistic f. I got that we can order each partition as: Each partition of n = {the empty partition} + 1(n-1 choose k numbers)(order the rest n-1-k numbers) + 1(n-2 choose k numbers)n(order n-k-2 numbers).....{123...n} , the last partition means that n appear in the last partition.
Is that right? I cannot think about something else
The statistics of frequency of the f-values you are looking for is sequence A133611 in the Online Encyclopedia of Integer Sequences (OEIS). You can find recurrence equations and generating functions there.