How many different patterns to transport load with these condition? (I am not expert in English. I will write as well as I can.)
To transport load, you have to put load into trailer cart and transport it in rounds. (I will improve this part (if I can. I am not expert in English.) but I think you can under stand the question from an example.)
There are 5 sizes of carts which are 5,4,3,2,1 unit.
Conditions:
1.Each round must have at least one cart, for example (total load is 2 unit)
Round 1: 1/5 1/4
(mean "Round 1 : Use 5-unit cart with 1 unit of load , Use 4-unit cart with 1 unit of load")
/ , count as a pattern
Round 1: 1/5     Round 2: 1/4
/ , count as a pattern
Round 1: 1/5 1/4     Round 2:
x , doesn't count as one pattern
2.Load in each cart must be positive integer, for example (total load is 2 unit)
Round 1: 1.5/5 0.5/4
x , doesn't count as a pattern
Round 1: 2/5 0/4
x   doesn't count as a pattern
3.Each round must't have 2 or more same size cart, for example (total load is 2 unit)
Round 1: 1/5 1/5
x , doesn't count as a pattern
4.Larger size cart must be in earier order, for example (total load is 2 unit)
Round 1: 1/4 1/5
x , doesn't count as a pattern
For total load is 5 unit, example of patterns
(1)
Round 1: 1/5 1/4
Round 2: 1/5 1/4
Round 3: 1/5
/ , count as one different pattern
(2)
Round 1: 1/5 1/4
Round 2: 2/5
Round 3: 1/5
/ , count as one different pattern
(3)
Round 1: 1/1
Round 2: 2/5
Round 3: 2/5
/ , count as one different pattern
(4)
Round 1: 3/5
Round 2: 2/5
/ , count as one different pattern
(5)
Round 1: 2/5
Round 2: 3/5
/ , count as one different pattern
...
If I give number of total load is 20 unit, how many patterns to transport load with these condition ?
(I think I don't forgot some condition.)
 A: I'll try to brute force it. Since there is no restriction on the number of the total rounds.

#twenty rounds
$$5^{20}$$
#nineteen rounds
1 round two cart + 1 cart two loads
$${19\choose1}5^{18}\left[{5\choose2}+4\right]$$
#eighteen rounds
one stack $(2,)$
(1,1,1)+(2,1)+(3,)
$${18\choose1}5^{17}\left[{5\choose3}+4\cdot4+3\right]$$
two stacks $(1,1)$
(2,)+(1,1)
$${18\choose2}5^{16}\left[{(4+{5\choose2})\cdot(4+{5\choose2})}\right]$$
... but I think we should use as small rounds as possible so I stop here.
# two rounds(least)
strategy: first round full load(5 carts, 15 loads) for every cart then consider second round
(5,)+(4,1)+(3,2)+(3,1,1)+(2,2,1)+(2,1,1,1)+(1,1,1,1,1)
$$1\cdot\left[1+\\2\cdot4+\\3\cdot3+\\3\cdot{4\choose2}+\\{4\choose2}\cdot3+\\{4\choose1}{4\choose3}+\\1\right],$$
which is 1+8+9+18+18+16+1=$71$.
For this question to be more realistic you should add more restrictions on so you won't waste resources, because for other cases the number will be, as I just show above, astronomical figures.
