Probability of split hands (infinite deck)

I want to work out the expected return for some split hands. Say you are permitted to split up to three times, so as to form four separate hands. The starting point should be what's the probability of having 2 hands, 3 hands or 4 hands to split, because from there you can work out expected returns.

Assuming an infinite deck, these are my results:
2 hands = 12/13 x 12/13 = 85.20710%
3 hands = 2x(1/13 x 12/13 x 12/13 x 12/13) = 12.10042%
4 hands = (1/13 x 1/13) + 2x(1/13 x 1/13 x 12/13) + 2x(1/13 x 1/13 x 12/13 x 12/13) = 2.69248%

Can any one confirm that's correct? For some reason from there I can't reconcile with the Wizard's Appendix 1 on infinite deck splitting expected returns:

I am only out by <0.1% but I want to get it spot on and understand the difference.

Thanks in advance, Wardos.

1 hand = 12/13
2 hands = 144/2197
3 hands = 3456/371293
4 hands = 769/371293
These match yours exacly if I multiply them by 13.

Thanks Miplet - I didn't have 1 hand probability in there because I had already assumed you had a splitting hand once dealt. Either way, our numbers agree. I have a feeling I am not matching up with the Wizard Appendix 1 splitting expected returns because I have used unrounded numbers throughout. Like I said the difference was very small.

With thanks
Wardos