Suppose we have list of Q integers $p_1, p_2,..., p_Q$. In round $k$ we have combinations of the integers. For example Q = 3, the combinations of $k$ round are:
k= 1, $p_1, p_2,p_3$ .
k= 2, $p_1p_1, p_1p_2,p_1p_3$,
$p_1p_2, p_2p_2,p_2 p_3$,
$p_1p_3, p_2p_3, p_3p_3$.
k = 3, $p_1p_1p_1, p_1p_1p_2,p_1p_1p_3$
$p_1p_2p_1, p_1p_2p_2,p_1p_2 p_3$
$p_1p_3p_1, p_1p_3p_2,p_1p_3p_3$,...,
$p_3p_3p_1,p_3p_3p_2,p_3p_3p_3$
k=4, $p_1p_1p_1p_1, p_1p_1p_1p_2,p_1p_1p_1p_3$,
$p_1p_1p_1p_2p_2, p_2p_1p_2p_2,p_1p_1p_2 p_3,$...,
$ p_1p_1p_3p_Q,p_1p_2p_3p_3, ...,p_3p_3p_3p_3$
How do I mathematically formulate this problem using kind of $\sum \prod$?
many thanks
I like
$$S(k)=\{p_{\sigma_1}p_{\sigma_2}\cdots p_{\sigma_k}:1\leq\sigma_1\leq\sigma_2\leq\cdots\leq \sigma_k\leq Q\}.$$