A135810 -id:A135810 - cp's OEIS frontend

A135809 Number of coincidence-free length n lists of 3-tuples with all numbers 1,...,n in tuple position k, for k=1,2,3.

1, 0, 7, 194, 13005, 1660964, 363083155, 125447139558, 64534483387801, 47199368682436040, 47309812970969661471, 63078455495155600593290, 109143265990975402533003877, 240033842542243124391262433004

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008, May 21 2008

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.

			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).

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=3.

G. C. Greubel, Table of n, a(n) for n = 0..181

Cf. A089041 (coincidence-free 2-tuples), A135810 (coincidence-free 4-tuples).

Table[Sum[(-1)^k Binomial[n, k](n-k)!^3, {k, 0, n}], {n, 0, 13}] (* Geoffrey Critzer, Jun 17 2013 *)

a(n)=sum(k=0,n,(-1)^k*binomial(n, k)*(n-k)!^3) \\ Charles R Greathouse IV, Nov 17 2016

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

A135811 Number of coincidence-free length n lists of 5-tuples with all numbers 1..n in tuple position k, for k=1..5.

Original entry on oeis.org

1, 0, 31, 7682, 7931709, 24843464324, 193342583284315, 3250662144028779654, 106536051676371091349113, 6291424280473807580386161416, 629175403160580417773688864819351, 101332539752812925263300043667901615370, 25215284734588360186537964188340607229390261

Offset: 0

Wolfdieter Lang, Jan 21 2008

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.

			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.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=5.

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

Cf. A135810 (coincidence-free 4-tuples). A135812 (coincidence-free 6-tuples).

Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^5, {k, 0, n}], {n,0,25}] (* G. C. Greubel, Nov 23 2016 *)

a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^5. See the Charalambides reference a(n)=B_{n,5}.

cp's OEIS Frontend

A135809 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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