I believe I have gotten all of the ways now - thanks for the hints below Yun, Andre Nicolas, and Gerry Myerson. If anyone could confirm my answer (I feel there should be more possibilities, but for the numbers to be increasing left-right, I have found them all I believe). Sorry I don't have the headings on this chart, but from left to right it is: Numbers 1-6 (one for each column) with the last column being "Sum," if it was not already obvious.
I can verify your solution. I have one more than you. I've also put up a list (everything is divided by 2). I also put it up in the order that mirrors the smaller example's list (at least, as I interpreted it - perhaps, putting the list in strictly number-increasing order would have been a good idea too).
1,1,1,1,1,8
1,1,1,1,2,7
1,1,1,1,3,6
1,1,1,2,2,6
1,1,1,1,4,5
1,1,1,2,3,5
1,1,2,2,2,5
1,1,1,2,4,4
1,1,1,3,3,4
1,1,2,2,3,4
1,2,2,2,2,4
1,1,2,3,3,3
1,2,2,2,3,3
2,2,2,2,2,3