[seqfan] nomenclature in Bower's transformations

[Richard J. Mathar]

I'm wondering what some sentences in Bower's combinatorics
explained in https://oeis.org/transforms2.html actually mean:

i) Does "The sequence a_n .. represents the number of colors of a box holding
  n balls can be"
  mean:
  "The sequence a_n.. represents the number of colors of a box
  holding n balls can have"

ii) What is meant by the size of a box? Is it the number of balls in it,
  or is it a sort of capacity (meaning the maximum number of balls in it)?

iii) What are the objects in "each object distinct" in the example of BGJ?
  Why doesn't it say "each box distinct by color and/or size"

The formula for EFJ_k in the "Algorithms" chapter means: 
sum the a_{p_i} multiplied by
the multinomial (p_1,p_2,...p_k | n). It basically means: distribute
the total number of n balls over k boxes of different color. 
[There are partitions of n where boxes are too small to keep
p_i balls, and these don't contribute to the sum via a_{p_i}=0.]
I don't see where the formula says that no two boxes are the same size,
as the partitions allows p_i=p_j for some pairs of i and j.
So isn't this actually a formula for EIJ?
[Or should the "enumerate the distinct partitions of n into k parts"
actually read "enumerate the partitions of n into k distinct parts"
three lines earlier?]

Is a factor 1/k! missing on the right hand side of this formula for EFJ_k ?