Every Deck

A deck ordering is a permutation, and a permutation can be written as a mixed-radix number. At each position we write down how many still-unplaced cards were smaller than the one we chose. That digit, times that position’s factorial weight, is its contribution. Sum them and you have the index.

The last digit is always 0 — by then one card is left, so there is nothing to choose between. That is why the effective radices run 52 down to 2, and why the space is exactly 52!.