If the player is immediately to the left of the dealer, I think the answer is:
52 x (52-p) x (52-2p) x ... x (52-(n-1)p),
and if the player is the kth player, I think the answer is:
(52-k+1) x (52-p-k+1) x (52-2p-k+1) x ... x (52-(n-1)p-k+1).
Am I right?
I'm assuming that the pack has 52 cards, and that dealing starts immediately to the left of the dealer and goes round clockwise.