A135813 Number of coincidence-free length n lists of 7-tuples with all numbers 1..n in tuple position k, for k=1..7.

1, 0, 127, 279554, 4585352445, 358295150440964, 100303980203191474555, 82605709118517742843295238, 173237539725464803175622157326841, 828591383820135935294977528049328110600, 8285921143688038883760817278192753357365412471

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy

Comments

a(n) enumerates (ordered) lists of n 7-tuples such that every number from 1 to n appears once at each of the seven tuple positions and the j-th list member is not the tuple (j,j,j,j,j,j,j), for every j=1..n. Called coincidence-free 7-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.

Examples

			7-tuple combinatorics: a(1)=0 because the only list of 7-tuples composed of 1 is [(1,1,1,1,1,1,1)] and this is a coincidence for j=1.
7-tuple combinatorics: from the 2^7=128 possible 7-tuples of numbers 1 and 2 all except (1,1,1,1,1,1,1) appear as first members of the length 2 lists. The second members are the 7-tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2) = 2^7-1 = 127 lists is [(1,1,1,1,1,1,2),(2,2,2,2,2,2,1)]. The list [(1,1,1,1,1,1,1),(2,2,2,2,2,2,2)] does not qualify because it has in fact two coincidences, those for j=1 and j=2.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=7.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..90

Crossrefs

Cf. A135812 (coincidence-free 6-tuples).

Programs

Mathematica

Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^7, {k, 0, n}], {n,0,25}] (* G. C. Greubel, Nov 23 2016 *)

Formula

a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^7. See the Charalambides reference a(n)=B_{n,7}.