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%I A135812 #19 Jul 01 2023 08:27:52
%S A135812 1,0,63,46466,190916733,2985028951044,139296156465612475,
%T A135812 16389185827288545027462,4296451238117542245438597369,
%U A135812 2283341354940565366869098996941832
%N A135812 Number of coincidence-free length n lists of 6-tuples with all numbers 1,...,n in tuple position k, for k=1..6.
%C A135812 a(n) enumerates (ordered) lists of n 6-tuples such that every number from 1 to n appears once at each of the six tuple positions and the j-th list member is not the tuple (j,j,j,j,j,j), for every j=1,..,n. Called coincidence-free 6-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
%D A135812 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=6.
%H A135812 G. C. Greubel, <a href="/A135812/b135812.txt">Table of n, a(n) for n = 0..100</a>
%F A135812 a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^6. See the Charalambides reference a(n)=B_{n,6}.
%e A135812 6-tuple combinatorics: a(1)=0 because the only list of 6-tuples composed of 1 is [(1,1,1,1,1,1)] and this is a coincidence for j=1.
%e A135812 6-tuple combinatorics: from the 2^6=64 possible 6-tuples of numbers 1 and 2 all except (1,1,1,1,1,1) appear as first members of the length 2 lists. The second members are the 6-tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^6-1 =63 lists is [(1,1,1,1,1,2),(2,2,2,2,2,1)]. The list [(1,1,1,1,1,1),(2,2,2,2,2,2) does not qualify because it has in fact two coincidences, those for j=1 and j=2.
%t A135812 Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^6, {k, 0, n}], {n,0,25}] (* _G. C. Greubel_, Nov 23 2016 *)
%Y A135812 Cf. A135811 (coincidence-free 5-tuples). A135813 (coincidence-free 7-tuples).
%K A135812 nonn,easy
%O A135812 0,3
%A A135812 _Wolfdieter Lang_, Jan 21 2008

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