<s>[INST] How many ways can you build a football/soccer team? I'm reading a book on the history of soccer. There are several ways to position a team, for example, four defenders, four midfielders and two forwards is the common 4-4-2. How many ways can you build a team, given that there are 10 players that move because the keeper is always in the same position?
I made some schemes:
10
1-9
1-1-8
1-1-1-7
1-1-1-1-6
1-1-1-1-1-5
1-1-1-1-1-1-4
1-1 1-1-1-1-1-3
1-1-1-1-1-1-1-1-2
1-1-1-1-1-1-1-1-1-1

1-2-7
1-2-1-6
1-2-1-1-5
1-2-1-1-1-4
1-2-1-1-1-1-3
1-2-1-1-1-1-1-2
1-21-1-1-1-1-1-1

1-3-6
1-3-1-5

Then I thought of another scheme, one that would group all possible combinations of diagrams from the number of lines that are willing players on the court.
L1 10
L2 1-9,9-1,2-8,8-2,3-7,7-3,6-4,4-6,5-5
L3 

1-1-8,1-8-1,8-1-1,1-2-7,1-7-2, 2-7-1,2-1-7,7-2-1, 7-1-2,1-3-6,1-6-3 .....
I can not find the formula to translate this, if it is correct.
In looking for an answer I found this site. Could you guide me to a book or article to solve my problem?
 [/INST] You can use a generating function to solve this question: $$(1+x+x^{2}+x^{3}+...+x^{10})(1+x^{2}+x^{4}+...+x^{10})(1+x^{3}+x^{6}+x^{9})(1+x^{4}+x^{8})(1+x^{5}+x^{10})(1+x^{6})(1+x^{7})(1+x^{8})(1+x^{9})(1+x^{10})$$
Expand the above polynomial and the coefficient of $x^{10}$ is the answer.
Mathematica code:
$$\text{Coefficient}\left[\text{Product}\left[\text{Sum}\left[x^{i*k},\left\{i,0,\text{Floor}\left[\frac{10}{k}\right]\right\}\right],\{k,1,10\}\right],x,10\right]$$
The answer is 42.
For more information, see Lectures on Integer Partitions and Integer partitions (Wikipedia).
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