Ordinal numbers I

Georg Cantor made many brillant discoveries, but the ones I find the most fascinating are in the realm of infinite sets, especially the Ordinal Numbers.

They are a new kind of numbers with some strange arithmetic rules :
$(\omega +1)+({\omega }^2+\omega \times 2+1)={\omega }^2+\omega \times 2+1$
$({\omega }^2+\omega \times 2+1)+(\omega +1)={\omega }^2+\omega \times 3+1$

Let the following ordinal numbers:
${\sigma }_1={\omega }^2$
${\sigma }_2=\omega +1$
${\sigma }_3=\omega \times 2+2$
${\sigma }_4=\omega \times 4+3$
${\sigma }_5=\omega \times 8+4$

If we add the five ${\sigma }_i$ ordinal numbers using all possible 120 permutations and sort them (using the ordinal number order)
- how many different sums can we obtain
- what is the $1^{st}$ (lowest) value
- what is the ${17}^{th}$ value

Answer format: comma delimited

The ordinal numbers are represented using the Cantor Normal Form

• the symbol $\omega$ is represented with a w
• exponentiation is represented with a ^
• the symbol $\times$ is represented with a .
• all spaces are removed

${\omega }^2+\omega \times 3+1\to$ w^2+w.3+1

[My timing : < 1 s]