# Summation of all j-combinations (Expanding composition formula)

I found a formula for a problem that I was trying to solve, the Formula 3.2 in Section 3 at page 441 of this document.I am a little unsure about the "Summation over all j-combinations". Here is what I think but I am unsure about it. For example for k=3

\begin{align*}F(n,k;h_1,p_1;h_2,p_2;h_3,p_3)&=\binom{n-h+k-1}{k-1}+(-1)^1\binom{n-h+k-j-1-(p_1-h_1)}{k-1}+(-1)^2\binom{n-h+k-j-1-(p_1-h_1)-(p_2-h_2)}{k-1}+(-1)^3\binom{n-h+k-j-1-(p_1-h_1)-(p_2-h_2)-(p_3-h_3)}{k-1} \end{align*}

Is this the correct expansion? According to a couple of numerical solutions that I have tried it works for me. But I just want to be sure before I actually use it.

Edit: So this is the correct expansion from what I understand? \begin{align*}F(n,k;h_1,p_1;h_2,p_2;h_3,p_3)=\binom{n-h+k-1}{k-1}+(-1)^1\Bigg(\binom{n-h+k-j-1-(p_1-h_1)}{k-1}+\binom{n-h+k-j-1-(p_2-h_2)}{k-1}+\binom{n-h+k-j-1-(p_3-h_3)}{k-1}\Bigg)+(-1)^2\Bigg(\binom{n-h-(p_1-h_1)-(p_2-h_2)}{k-1}+\binom{n-h-(p_2-h_2)-(p_3-h_3)}{k-1}+\binom{n-h-(p_1-h_1)-(p_3-h_3)}{k-1}\Bigg)+(-1)^3\binom{n-h+k-j-1-(p_1-h_1)-(p_2-h_2)-(p_3-h_3)}{k-1} \end{align*}

Since $k=3$, there are $\binom31=1$ $1$-combinations: $\{1\}$, $\{2\}$, and $\{3\}$. There are $\binom32=3$ $2$-combinations: $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$. And there is $\binom33=1$ $3$-combination, $\{1,2,3\}$. For $j=2$, for instance, $k-j-1=0$, so the $\Sigma^*$ expression expands to
\begin{align*} \binom{n-h-(p_1-h_1)-(p_2-h_2)}2&+\binom{n-h-(p_1-h_1)-(p_3-h_3)}2\\ &+\binom{n-h-(p_2-h_2)-(p_3-h_3)}2\;, \end{align*}
where the first term is for $\{1,2\}$, the second for $\{1,3\}$, and the third for $\{2,3\}$.
The $j=1$ expansion will also have three terms, while the $j=3$ expansion will have only one.