Solitaire all cards dealt face up in 13 columns, then 12, then 11, 10, 9
It is not exactly the same rules but it is very similar to Perseverance 


It's a solitaire game starting with 12 columns, not 13 (the 4 aces are moved to the foundations at the beginning)
You have to build the foundation from ace to king
you can only stack cards of the same suit in the columns
when you are stuck you pile the columns without shuffling and re-deal in less columns. Not specifically 13, 12, 11, 10, 9 as you mentioned but in pile of 4 cards.