2810077

How many functions are there from 9 to 7 if every image in the codomain has 3 arguments in the domain?

How many functions are there from [9 to [7] if every image in the codomain has 3 arguments in the domain?]

The general species associated to this sort of problem is :

$$E(E(X) \cdot Y) \cdot E(Y)$$

We consider objects like {1,2,3}-A, {4,5,6}-B, {7,8,9}-C and {D,E,F,G}

Overall we get a formula for specific cardinality :

$$E_3(E_3(X) \cdot Y)\cdot E_4(Y) $$

Passing to e.g.f. one has

$$ \frac {1} {3!} \left( \frac {x^3}{3!} y \right)^3. \frac {y^4} {4!} $$

after calculus we got the sollution that is the coefficient of the e.g.f : $$ 58880 .\frac {x^9}{9!}. \frac {y^7}{7!} $$