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A135810 Number of coincidence-free length n lists of 4-tuples with all numbers 1..n in tuple position k, for k=1..4.

%I A135810 #26 Jan 29 2025 06:58:01
%S A135810 1,0,15,1250,326685,205713924,267499350955,643364455632870,
%T A135810 2637753876195952185,17316358344270678304520,
%U A135810 173227930768100416550798151,2536860701329458663625695526890
%N A135810 Number of coincidence-free length n lists of 4-tuples with all numbers 1..n in tuple position k, for k=1..4.
%C A135810 a(n) enumerates (ordered) lists of n 4-tuples such that every number from 1 to n appears once at each of the four tuple positions and the j-th list member is not the tuple (j,j,j,j), for every j=1..n. Called coincidence-free 4-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
%D A135810 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=4.
%H A135810 G. C. Greubel, <a href="/A135810/b135810.txt">Table of n, a(n) for n = 0..143</a>
%F A135810 a(n) = Sum_{j=0..n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^4 ). See the Charalambides reference a(n) = B_{n,4}.
%e A135810 4-tuple combinatorics: a(1)=0 because the only list of 4-tuples composed of 1 is [(1,1,1,1)] and this is a coincidence for j=1.
%e A135810 4-tuple combinatorics: from the 2^4 possible 4-tuples of numbers 1 and 2 all except (1,1,1,1) appear as first members of the length 2 lists. The second members are the 4-tuples obtained by interchanging 1 and 2. E.g., one of the a(2)=15 lists is [(1,1,1,2),(2,2,2,1)]. The list [(1,1,1,1),(2,2,2,2)] does not qualify because it has in fact two coincidences, those for j=1 and j=2.
%t A135810 Table[Sum[(-1)^k Binomial[n,k](n-k)!^4,{k,0,n}],{n,0,11}] (* _Geoffrey Critzer_, Jun 17 2013 *)
%Y A135810 Cf. A135809 (coincidence-free 3-tuples), A135811 (coincidence-free 5-tuples).
%K A135810 nonn,easy
%O A135810 0,3
%A A135810 _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008