Counting complexity of a scheduling problem.
Edit: It seems that the questioner knew that this was an open problem: https://mathoverflow.net/questions/33796/existence-of-a-pseudo-polynomial-time-algorithm-for-a-counting-problem.  For what it is worth, it took me some time to find the paper, and I think that the time was wasted.



This question is stated as an open problem in [Ber09], where the same problem except that the numbers in the input are written in binary is shown to be #P-complete by a reduction from the counting version of the Subset Sum problem.

References

[Ber09] Gerardo Berbeglia.  The counting complexity of a simple scheduling problem.  Operations Research Letters, 37(5):365–367, Sept. 2009.  http://dx.doi.org/10.1016/j.orl.2009.05.004