The number of rook paths on an MxN sized grid is called W(M,N) by Steven Finch. I will adopt this notion here.
The tables for W(3,*) up to W(12,*) were
computed at 9-Jul-1994 by me (Heiner Marxen)
on an Apollo HP9000-710.
Computation stopped, when the number length exceeded 63 decimal
places.
In brackets is given floor(log10(W)) (i.e. length-1).
Up to W(13,13)
the results have been confirmed by
Andreas Gammel.
The table for W(13,*) has been computed on an SGI IRIX workstation (R4400)
at 20-May-1995.
The table for W(14,*) and W(15,*) have been computed on an HP9000/819
at 10-Sep-1996 and 11-Sep-1996.
W(1,N) is not given, it is proven to be 1.
W(2,N) is not given, it is proven to be
2(N-1).
Some timing info:
W( 3,*) < 1 sec
W( 4,*) < 1 sec
W( 5,*) < 1 sec
W( 6,*) < 1 sec
W( 7,*) < 1 sec
W( 8,*) < 3 sec
W( 9,*) < 11 sec
W(10,*) < 51 sec, using 0.7 MB
W(11,*) < 4.5 min, using 3.2 MB
W(12,*) < 28 min, using 14.7 MB
W(13,*) < 103 min, using 4.45 MB
W(14,*) < 211 min, using 12.09 MB
W(15,*) ca. 16.5 hours, using 33.36 MB