Given d dice each with n sides, find the probability that when they are rolled at once, there are no two consecutive numbers.

This task may be quite difficult. For a warm-up, try finding the numerators for fixed d such as {1,2,3} or for fixed n such as {2,3,4}.

Note 1: I don't have a formula so much as an algorithm.
Note 2: This problem arose as an attempt to solve http://perplexus.info/show.php?pid=12342 by Larry which uses non-independent cards instead of dice.

(In reply to Neatening the table by Charlie)

I took the main diagonal of the numerators 2,9,46,335,2786 and went over to the OEIS.  There was exactly one match:  http://oeis.org/A219614 "Number of ways to put n labeled objects into n labeled boxes so that no two nonempty boxes are adjacent."

Posted by Brian Smith on 2021-01-28 10:30:21