Find generating function for $x_1+x_2+x_3+x_4=n$ while that $x_1\ne x_2$ and $x_1\lt x_2$

Question

Find generating function for $x_1+x_2+x_3+x_4=n$ with the following constraints:

a) $x_1\ne x_2$

b) $x_1\lt x_2$

Can you please give me a direction or hint of how to represent in a generating function the fact that $x_1$ differs from $x_2$?

Answer 1

Hint: if you try to find the number of compositions, satisfying the above criteria, you can do it in 3 steps.

1. Calculate number of compositions for $x_1+x_2+x_3+x_4=n$ unrestricted, call it $M_1$.

2. Calculate number of compositions for $2x_1+x_3+x_4=n$, unrestricted, call it $M_2$.

3. $M_1 -M_2$ will give you the number of compositions without $x_1 < x_2$ restriction. To sort this out, take $\frac{M_1 -M_2}{2}$ since half of $M_1 -M_2$ will be with $x_1<x_2$ and half with $x_1 > x_2$.

Obviously, cases $1$ and $2$ will use much simpler generating functions.