W. Lang, Dec 21, 2007 a(n,m) tabl head (triangle) for A135814 (number of coincidence free m-tupel with all numbers 1,...,n-m distributed over all tuple positions). n\m 0 1 2 3 4 5 6 7 8 9 ... 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 3 0 1 0 1 0 0 0 0 0 0 4 0 2 3 0 1 0 0 0 0 0 5 0 9 26 7 0 1 0 0 0 0 6 0 44 453 194 15 0 1 0 0 0 7 0 265 11844 13005 1250 31 0 1 0 0 8 0 1854 439975 1660964 326685 7682 63 0 1 0 9 0 14833 22056222 363083155 205713924 7931709 46466 127 0 1 . . . The second column gives the subfactorials (derangements) A000166 = [1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, ...]. The columns give: A000007 (m=0), A000166 (subfactorials), A089041, A135809 - A135813, for m=0..7. The row sums give A135815 = [1, 1, 1, 2, 6, 43, 707, 26396, 2437224, 598846437,...]. The alternating row sums give A135816 = [1,-1,1,-2,2,9,231,-208,-903776,-143213213,...]. Example: a(5,2)=26 from the number of coincidence free length 3 =n-m lists of (m=2)-tuples composed of the numbers 1,2 and 3. E.g. [(1,2),(2,1),(3,3)] does not qualify because of the concidence in the j=3 position. [(1,2),(2,2),(3,1)] has coincidence for j=2. [(1,3),(2,2),(3,1)] qualifies. [(1,1),(2,2),(3,3)] has three coincidences, for j=1,2 and 3, hence does not qualify. ###################################### e.o.f. #########################################################