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A135811
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Number of coincidence-free length n lists of 5-tuples with all numbers 1,...,n in tuple position k, for k=1..5.
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3
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1, 0, 31, 7682, 7931709, 24843464324, 193342583284315, 3250662144028779654, 106536051676371091349113, 6291424280473807580386161416, 629175403160580417773688864819351
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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 5-tuples such that every number from 1 to n appears once at each of the five tuple positions and the j-th list member is not the tuple (j,j,j,j,j), for every j=1,..,n. Called coincidence-free 5-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=5.
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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!)^5 ). See the Charalambides reference a(n)=B_{n,5}.
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EXAMPLE
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5-tuple combinatorics: a(1)=0 because the only list of 5-tuples composed of 1 is [(1,1,1,1,1)] and this is a coincidence for j=1.
5-tuple combinatorics: from the 2^5 possible 5-tuples of numbers 1 and 2 all except (1,1,1,1,1) appear as first members of the length 2 lists. The second members are the 5-tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^5-1 =31 lists is [(1,1,1,1,2),(2,2,2,2,1)]. The list [(1,1,1,1,1),(2,2,2,2,2) does not qualify because it has in fact two coincidences, those for j=1 and j=2.
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MATHEMATICA
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Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^5, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Nov 23 2016 *)
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CROSSREFS
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Cf. A135810 (coincidence-free 4-tuples). A135812 (coincidence-free 6-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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