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A135809
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Number of coincidence-free length n lists of 3-tuples with all numbers 1,...,n in tuple position k, for k=1,2,3.
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3
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1, 0, 7, 194, 13005, 1660964, 363083155, 125447139558, 64534483387801, 47199368682436040, 47309812970969661471, 63078455495155600593290, 109143265990975402533003877, 240033842542243124391262433004
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OFFSET
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0,3
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COMMENTS
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a(n) enumerates (ordered) lists of n three-tuples such that every number from 1 to n appears once at each of the three tuple positions and the j-th list member is not the tuple (j,j,j), for every j=1,..,n. Called coincidence-free 3-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
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REFERENCES
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Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=3.
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LINKS
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FORMULA
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a(n) = Sum_{j=0,..,n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^3 ). See the Charalambides reference a(n) = B_{n,3}.
a(n) ~ n!^3 ~ (2*Pi)^(3/2) * n^(3*n + 3/2) / exp(3*n). - Vaclav Kotesovec, Nov 19 2016
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EXAMPLE
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3-tuple combinatorics: a(1)=0 because the only list of 3-tuples with numbers 1 is [(1,1,1)] and this is a coincidence for j=1.
3-tuple combinatorics: the a(2)=7 coincidence-free 3-tuple lists of length n=2 are [(1,1,2),(2,2,1)], [(1,2,1),(2,1,2)], [(2,1,1),(1,2,2)], [(2,2,1), (1,1,2)], [(2,1,2),(1,2,1)], [(1,2,2),(2,1,1)] and [(2,2,2),(1,1, 1)]. The list [(1,1,1),(2,2,2)] has in fact two coincidences (j=1 and j=2).
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MATHEMATICA
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Table[Sum[(-1)^k Binomial[n, k](n-k)!^3, {k, 0, n}], {n, 0, 13}] (* Geoffrey Critzer, Jun 17 2013 *)
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PROG
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CROSSREFS
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Cf. A089041 (coincidence-free 2-tuples), A135810 (coincidence-free 4-tuples).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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