Let [n] denote an integer partition of n. Let S([n]) be the representation of 
[n] by its parts p(i)… and their multiplicities mult(p(i)). E.g. 
S([1,2,2,2,4])=[1^1,2^3,4^1] is a partition of n=11. According to the partition 
[n] n unlabelled elements are distributed into the parts of the partition. E.g. 
for n=4 and its partition [4] all elements are within the part p(1)=4 and for 
n=4 and its partition [1,3] one element belongs to part p(1)=1 and three to the 
remaining part p(2)=3. Now, these parts (we can speak of “goups” also) are 
treated as labelled. E.g. for n=4 and its partition [2,2] we have the two 
labelled parts 2_1 and 2_2. These labelled parts are distributed into the 
labelled boxes with empty boxes forbidden. For [2,2] we end up with two 
possibilities (<…> indicates a box): <2_1>,<2_2> and <2_2>,<2_1>. The 
possibilities for each partition of n are summed up to yield L([n],m).

Let L([n],m) be the number of all possible distributions of the labelled parts 
of specification S([n])=[1^mult(1),2^mult(2),3^mult(3),…,n^mult(n)] into m 
labelled boxes (also called parcels) with empty boxes excluded. In the following
 “prod_{i=1, p(i), [n]}^{n}” means “product over all parts p(i) with i=1,..,n of 
the single partition [n]. The sum over all P(n) integer partitions of n is 
written as “sum_{h=1,[n]}^P(n)”. Furthermore, S2(n,k) is the Stirling number of 
the second kind. Then 

L([n],m) = sum_{h=1,[n]}^P(n)  prod_{i=1, p(i), [n]}^{n} m! S2(p(i), m)^mult(i)

Observe that for any partition [n] which has a part p(i) < m this partition does
 not contribute to the sum L([n],m). A partition can contribute only if for all 
its parts p(i) >= m. To understand this fact we consider n represent n 
unlabelled elements. Each part p(i) of any integer partition [n] needs to 
contribute at least one unlabelled element to each of the m labelled boxes. 
This request expresses the symmetry among the n unlabelled elements.
%e A139359 Let <…> indicate a box. We illustrate the formula for n=4 where we have five 
integer partitions. For n=4 and m=2 we have 16 cases:

For specification S[4^1] which corresponds to the partition [4] we get
<a1,a2,a3>, <a4>
<a1,a2,a4>, <a3>
<a1,a3,a4>, <a2>
<a2,a3,a4>, <a1>
<a4>, <a1,a2,a3>
<a3>, <a1,a2,a4>
<a2>, <a1,a3,a4>
<a1>, <a2,a3,a4>
<a1,a2>, <a3,a4>
<a1,a3>, <a2,a4>
<a1,a4>, <a2,a3> 
<a2,a3>, <a1,a4>
<a2,a4>, <a1,a3> 
<a3,a4>, <a1,a2>

For specification S[2^2] which corresponds to the partition [2,2] we get
<a1>, <a2>
<a2>, <a1>

The last two lines make the difference to A019538(n=4,m=2)=14.

The other three partitions do not contribute for m=2, because they contain parts
 with p(i)<m. E.g. for n=4, m=2 and [1,3] the part p(1)=1 contributes a zero to 
the product and thus the product vanishes.