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A135812 Number of coincidence-free length n lists of 6-tuples with all numbers 1,...,n in tuple position k, for k=1..6. 3
1, 0, 63, 46466, 190916733, 2985028951044, 139296156465612475, 16389185827288545027462, 4296451238117542245438597369, 2283341354940565366869098996941832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

REFERENCES

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=6.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..100

FORMULA

a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^6. See the Charalambides reference a(n)=B_{n,6}.

EXAMPLE

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.

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.

MATHEMATICA

Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^6, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Nov 23 2016 *)

CROSSREFS

Cf. A135811 (coincidence-free 5-tuples). A135813 (coincidence-free 7-tuples).

Sequence in context: A212859 A239672 A136677 * A069452 A229846 A230674

Adjacent sequences:  A135809 A135810 A135811 * A135813 A135814 A135815

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang Jan 21 2008

STATUS

approved

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Last modified October 18 23:35 EDT 2021. Contains 348071 sequences. (Running on oeis4.)