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

1, 0, 15, 1250, 326685, 205713924, 267499350955, 643364455632870, 2637753876195952185, 17316358344270678304520, 173227930768100416550798151, 2536860701329458663625695526890

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
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.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=4.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..143

Crossrefs

Cf. A135809 (coincidence-free 3-tuples), A135811 (coincidence-free 5-tuples).

Programs

Mathematica

Table[Sum[(-1)^k Binomial[n,k](n-k)!^4,{k,0,n}],{n,0,11}] (* Geoffrey Critzer, Jun 17 2013 *)

Formula

a(n) = Sum_{j=0..n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^4 ). See the Charalambides reference a(n) = B_{n,4}.